The College Panda

SAT Math
Advanc& Cuide ard Workbook

For The
New SAT

Nielson Phu
 The College Panda

SAT Math
Advanced Guide and Workbook

Copyright @ 2015 The College Panda

All rights reserved.

ISBN: 978-0-9894964-2-1

No part of this book may be reproduced without written permission from the author.

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SAT & IG BOOK STORE 01227746409 1

 To Mom and Dad

SAT & IG BOOK STORE 01227746409 2

Introduction
The best way to do well on any test is to be experienced with the material. Nowhere is this more true than on
the SAT, which is standardized to repeat the same question types again and again. The purpose of this book is
to teach you the concepts and battle-tested approaches you need to know for all these questions types. If it's
not in this book, it's not on the test. The goal is for every SAT question to be a simple reflex, something you
know how to handle instinctively because you've seen it so many times before.
You won't find any cheap tricks in this book, simply because there aren't any that work consistently. Don't buy
into the idea that you can improve your score significantly without hard work.

Format of the Test

There are two math sections on the SAT. The first contains 20 questions to be done in 25 minutes without a
calculator. The second contains 38 questions to be done in 55 minutes and a calculator is permitted.

Some topics only show up in the calculator section. I've made sure to accurately divide the practice questions
into non-calculator and calculator components.

How to Read this Book

For a complete understanding, this book is best read from beginning to end. That being said, each chapter was
written to be independent of the others as much as possible. you may already be proficient in some
After all,

topics yet weak in others. If so, feel free to jump around, focusing on the chapters that are most relevant to

your improvement.
All chapters come with exercises. Do them. You won't master the material until you think through the
questions yourself.

About the Author

Nielson Phu graduated from New York University, where he studied actuarial science. He has obtained perfect
scores on the SAT and on the SAT math subject test. As a teacher, he has helped hundreds of students
throughout Boston and Hong Kong perform better on standardized tests. Although he continues to pursue
his interests in education, he is now an engineer in the Boston area.

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Table of Contents

1 Exponents & Radicals 7

Laws of exponents
Evaluating expressions with exponents
Solving equations with exponents
Simplifying square roots

2 Percent 15
Percent change
Simple interest and compound interest
Percent word problems

3 Exponential vs. Linear Growth 24

Linear growth and decay
Exponential growth and decay
Positive and negative association

4 Proportion 30
5 Rates 34
Conversion factors

6 Expressions 42
Combining like terms
Expansion and factoring
Combining, dividing, and splitting fractions

7 Manipulating & Solving Equations 49

Common mistakes to avoid
Tools for isolating variables
How to deal with complicated equations
8 More Equation Solving Strategies 62
Matching coefficients
Clearing denominators

9 Systems of Equations
Substitution
Elimination
Systems with no solutions and infinite solutions

Word problems
More complex systems
Graphs of systems of equations

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10 Inequalities 79
How to solve inequalities
Inequalityword problems
Graphs of inequalities

11 Word Problems 90
12 Lines 99
Slope and y-intercept
Equations of lines: slope-intercept form and point-slope form
Finding the intersection of two lines
Parallel and perpendicular lines
Horizontal and vertical lines

13 Interpreting Linear Models 109

14 Functions 115
What is a function?
When is a function undefined?
Composite functions
Finding the solutions to a function
Identifying function graphs

15 Quadratics 126
Tactics for finding the roots

Completing the square

The vertex and vertex form
The discriminant
Quadratic models

16 Synthetic Division 141

Performing synthetic division
Equivalent expressions
The remainder theorem

17 Complex Numbers 150

18 Absolute Value 154
19 Angles 160
Exterior angle theorem
Parallel lines

Polygons

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20 Triangles 168
Isosceles and equilateral triangles
Right triangles
Special right triangles
Similar triangles
Radians

21 Circles 189
Area and circumference
Arc length
Area of a sector
Central and inscribed angles
Equations of circles

22 Trigonometry 198
Sine, cosine, and tangent
Trigonometric identities
Evaluating trigonometric expressions

23 Reading Data 205

24 Probability 214
25 Statistics I 223
Mean, median, and mode
Range and standard deviation
Histograms and dot plots
Word problems involving averages

26 Statistics II 233
sampling
Statistical
Using and interpreting the line of best fit
Margin of error
Confidence intervals
Experimental design and conclusions

27 Volume 246
28 Answers to the Exercises 251

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 Exponents & Radicals
Here are the laws of exponents you should know:

Law Example

30

xm • xli 34 • 35

m—n
33

mn
(32)4

(xy)m x my m (2-3)3 23 • 33

33

34

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CHAPTER 1 EXPONENTS & MDICALS

Many students don't know the difference between

(—3)2 and — 32

Order of operations (PEMDAS) dictates that parentheses take precedence. So,

Without parentheses, exponents take precedence:

-32

The negative is not applied until the exponent operation is carried through. Make sure you understand this so
you don't make this common mistake. Sometimes, the result turns out to be the same, as in:

(—2)3 and 23

Make sure you see why they yield the same result.

EXERCISE 1: Evaluate WITHOUT a calculator. Answers for this chapter start on page 251.

1.
(—1)4 19. 50

2. 11. 20. 32

3. (—1)10 12. 21.

4. (—1)15 13. 23 x 32 x (-1)5 22. 53

5. 14. (-1)4 x 33 x 22 23. 5 3

(-1)8

6. 18 15. (-2)3 x (-3)4 24. 7

7. 16. 30 25. 7 2

17. 26. 103

9. -33 18. 4-1 27. 10 3

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EXERCISE 2: Simplify so that your answer contains only positive exponents. Do NOT use a calculator. The
first two have been done for you. Answers for this chapter start on page 251.

11. (x2y 3 —1 •a
1. 3x2 2x3 6x5 1 20. (a

8
2. 2k-4 4k2 •
6u4 21. . (b3)2
12.
8112

(m2n)3
3. 5x4 3x -2 22.
2uv2
•

13. (mn2)2

4. 7m3 —3m¯3 •

x2 1
14. 23.
x —3 x —2
5. (2x2) 3

3x4
mn
6. 3a2b—3 3 a-5b8 .
24.
15. m2n3
-2 2
(x
7
311
3
25.
6113
16. k-3
1

8. 3
(a2b3)2 m2
17. x2 x3 26.
n3
•

4
9.
xy
x3y2 18. (x 2X3
x2y3z4
19. (2m)2 • (3m3)2 x-3y-4z-5
10

ÄÅM LE then what:is thevålüe€of 3x in termsofy?

Let's avoid the trouble of finding what x is. Here we notice that the 2 in the exponent is the only difference
between the given equation and what we want. So using our laws of exponents, let's extract the 2 out:

31+2 32 y

31 Y
9

The answer is (D) .

ÉXAMPLE 2: If 3+Ø+?jwKatisffe altleofa?

Here we see that the bases are the same. The exponents must therefore be equal.

2a 6

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CHAPTER 1 EXPONENTS & RADICALS

Realize that 4 is just 22.

(22)a 22a 4
16
2b 2b 2b

Square roots are just fractional exponents:

1

But what about x3? The 2 on top means to square x. The 3 on the bottom means to cube root it:

We can see this more clearly if we break it down:

2 3
(x2)å

The order in which we do the squaring and the cube-rooting doesn't matter.

The end result just looks prettier with the cube root on the outside. That way, we don't need the parentheses.

EXAMPLE 4: mfo/lowingis
A) x Ox$—x4

1
The fourth root equates to a fractional exponent of so
4'

4 5

Answer (C) .

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The SAT will also test you on simplifying square roots (also called "surds"). To simplify a square root, factor
the number inside the square root and take out any pairs:

In the example above, we take a 2 out for the first Ü]. Then we take another 2 out for the second pair
Finally, we multiply the two 2's outside the square root to get 4. Of course, a quicker route would have looked
like this:

48

Here's one more example:

72-

To go backwards, take the number outside and put it back under the square root as a pair:

72

EXÅMPLE& if 4vG the valueof x?

B) j? 9.16 D)48

Solution 1: Moving the 4 back inside, we get

3-4

Now equating the stuff inside the square roots,

48 3x

48

16 x

Answer (C) .

Solution 2: Square both sides:

(4 v6)2 = ( 3x)2
16-3
16 — x

11

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CHAPTER 1 EXPONENTS & RADICALS

EXERCISE 3: Simplify the radicals or solve for x. Do NOT use a calculator. Answers for this chapter start on
page 251.

1. 12 10. 128

2. 96 11. 5Vä

12.

13.

14. 4

6. 3vfiS 15.

16. -

8. 200 17.

18. xv/f 216

12

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CHAPTER EXERCISE: Answers for this chapter start on page 251.

A calculator should NOT be used on the

following questions.
— 10, what is the value of 31¯3?

10

If a 3, what is the value of a? 3

10
9
1
10
9
1

3
10

If x2y3 10 and x3y2 8, what is the value of

Let n = 1
2
+ 14 16 + 18 150
x5y5?
What is the value of n? A) 18
A) 10 B) 20

B) 20 C) 40
C) 25 D) 80
D) 30

If a and b are positive even integers, which of the

If 42n+3 811+5, what is the value of n? following is greatest?

A) (-2a)b
B) (—2a)2b
C) (2a)b
D) 2112b

If 23, then x must equal Which of the following is equivalent to x T , for

all values of x?

A) b ax2

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CHAPTER 1 EXPONENTS & RADICALS

13

If x2 y3, for what value of z does x If xac • xbc x30, x > 1, and a -h b 5, what is
the value of c?

D) 10

10

If 2x+3 2X k(2X), what is the value of k? A calculator is allowed on the following

questions.

If 113 x and n 4 20x, where n > 0, what is the

value of x?

11

If xa, then what is the value of a?

2
3
15
4

If 333 and x7y6 3, what is the value of

4
3

12

2 x+2—3v6
If x > 0 in the equation above, what is the value
Of X?

A) 2.5

C) 3.5

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 Percent
EXAMPLE I; Jacob got 50% of'thequestions correct on a 30-questionåestYénd€90% on a 50 guestion test,
What percenf of au quesiions didjacob correct'
: Bae. a ± aa

First, let's find the total number of questions he got correct:

1
x 30 x 30 — 15
2

9
900/0 x 50 x 50 = 45
10

60 3
So he got 15 + 45 60 questions correct out of a total of 30 + 50 80 questions. 75
80 4

EXAMPLE 2: Arécora of dr1vmgv101ations by gype below.

101atiémT
din Sto Sign Parkin& Total
Truck .39 17 124
Car 160
'Total 1 43

owinglsclosest€öthepercentofgecordedp kiilgviolationsthatwerecommitted
by trucks?
A) 60/0

PART 'If tQata were yseå drivinv violation inKOrmation about 2,000 totaLviolations{in a
ståtg, which-of the followitqg iS the bestestimate Of thenumber of Speeding violations committed
by qrsinthepte
4)479 B)585 01063 D)1059

15

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CHAPTER 2 PERCENT

Part 1 Solution:
Truck Parking Violations 17
— 0.3953 400/0
Total Parking Violations 43

Answer (C) .

Part 2 Solution: The SAT will often ask you to estimate or predict certain information based on a smaller
sample size. In these questions, take what you learn from the smaller size and simply apply it to the larger
Car Speeding Violations 83
population. From the sample size, Now we can apply this same proportion
Total Parking Violations 284
to the state total of 2,000:
83
x 2000 585

Answer (B) .

EXAMPLYES:iTheprice Of adres$ then decr•edOb.r 250/6,

Here's the technique for dealing with these "series of percent change" questions. Let the original price be p.

When p is increased by 20%, you multiply by 1.20 because it's the original price plus 20%. When it's decreased
by 40%, you multiply by .60 because 60% is what's left after you take away 40%. Our final price is then

p x 1.20 x .60 x 1.25 — .90p

The final price is of the original price.

Example 3 shows the MOST IMPORTANT percent concept by far on the SAT. Never ever calculate the prices
at each step. String all the changes together to get the end result.

important to know why this works. Imagine again that the original price
It's is p and we want to increase it by
20%. Normally, we would just take p and add 20% of it on top:

p —I— .20p

But realize that

p —I— .20p — p (I + .20) ¯ 1.20p

And now we want to decrease this new price by 40%:

1.20p — .40) = (1.20) (0.60)p

which proves we can calculate the final price directly through this technique. Now we're set up to tackle the
inevitable compound interest questions on the SAT.

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EXAMPLE 4: Jonashasa sayings account$hatzearns cogipoundegpnnuplly. Heginiti91

deposit Vas $100(). •Whic\v the$0110wing expressiOn$ gives Of the ad@unt aftet JO year$t—n

B) 1000+3000)

A 3 percent interest rate compounded annually means he earns 3 percent on the account once a year. Keep in
mind that this isn't just 3% on the original amount of $1000. This is 3% of whatever's in the account at the time,
including any interest that he's already earned in previous years. This is the meaning of compound interest.
So if we're in year 5, he would earn 3% on the original $1000 and 3% on the total interest deposited in years 1
through 4.
If we try to calculate the total after each and every year, this problem would take forever. Let's take what we
learned from Example 3 and apply it here:

Year 1 total: 1000(1.03)

Year 2 total: 1000(1.03) (1.03)
Year 3 total: 1000(1.03) (1.03) (1.03) =
Year 4 total: —

See the pattern? Each year an increase of 3% so it's just 1.03 times whatever the value was last year. Note
is

that we're not doing any calculations out. Think of it as the price of a dress being increased by 3% ten times.

Therefore, the Year 10 total is answer (D)

of these;comßOund interes questions can be modeled by the equation i)! , wilere 4is
the totalmqmoufitaqcumulated, is thé prmci al oc the initial is the interesf@tet.nd:! is the
rugnbér Of timé$ intere9t is received.

EXAMPLE S: Jay puts an initial depesi€ of $100 a bank accoun€that earnsS percenV Interest each,
year, compoundeasemiann@ålly.EWh1ch -of tfref6110wiHb equations givesthe total doliÅr amount/A; in
the accou»t after -t years?

4090±9.10

The interest is compounded semiannually. That means twice a year. So interest is received 2t times. However,
we don't receive a full 5% each time interest The 5% interest rate is a yearly figure. We have to
is received.

divide it by 2 to get the semiannual rate: 2.5%. The answer is (D) .

Note that semiannual compounding is better than annual compounding. Why? With annual compounding,
you just get 5% on the initial amount after one year. That's just like 2.5% on the initial amount and then another
2.5% on the initial amount. But with semiannual compounding, you get 2.5% on the initial amount and then
you get 2.5% on the mid-year amount, which is greater than the initial amount because it includes the first
interest payment. Because you've already earned interest before the end of the year, you get a little extra.
This might not seem like a lot, but over many years, it can make a huge difference. The more times interest is
compounded, the more money you accumulate.

17

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CHAPTER 2 PERCENT

Ginteresti#ompoühdedrnOre year;thé PieviouStormula anbegeneralizedto

•where-A is theototalamount åccumulated, P is the principal O? mitial amount, r is the interest rate t isthe—•
Of the number Of times the interest iS compounded eachyear. You don'! need to •z
theSe$OÉfr1UI@Soif the math

Now we've shown you how to handle compound interest questions, let's take a step back to bring up
that
simple While compound interest lets you earn interest on interest you've earned, simple interest
interest.
means you get the same amount each time. Interest is earned only on the original amount, not on any interest
you've earned.

EXAMPLE 6'. investor decides to Offer bUsmess Ownei å$20,0010an at Omp e mterestof 5%per.z
year. Whicfrof gives dollarsnthemveStor will receive when
theåoaiiß repaid after t years?
—

At a simple interest of 5%, the investor will receive 20, 000(0.05) in interest each year. That amount does not
change because the 5% always applies to the original $20,000 under simple interest. So after t years, he will
receive a total of 20, 000 (0.05)t in interest.

The amount he will be repaid after t years is then

A Original amount + Total interest

20, OOO + 20,
— 20, ooo(l + 0.050

Notice how we factored out the 20,000 in the last step. The answer is (D) . The answer would have been (A)
under compound interest.

Fof simple formulmis

P(Y40t)
whé!eu4 is the accumulatecp is the ihieréÅt
istiJénumberoftimesinterestisearned ( icallythenumber of years);

EXAMPLE74 firs year, the chickens onafarm laid 30% less €ygs aid VaStyeaw Ifthey laid 3,300
v.ggsthisye 'how many did they iåy lasfyear?

This Year (.70) (Last Year)

3,500 Year)

5,000 Last Year

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percent change (a.k@Pereent increase/deerease) iS calculateda$follow$,

new value •old value

0(osGange

For example, if the price of a dress starts out at 80 dollars and rises to 90 dollars, the percent change is:

90 80
x 100 — 12.5%
80

If percent change is positive, it's a percent increase. Negative? Percent decrease. It's important to remember
that percent change is always based on the original value.

EXAMPLE Ina particular sto e, the nugvber Of TVS sold tthe week df Blaa Friday was •68ü The number
6f,TVS$old the following week was 500. TV sales theweek following Black Friday werewhat percentless
thanTV Salesthe Week ofoElaclf Friday (roundedto•the nearestrcent)?

500 - 685
-0.27
685

We put the difference over 685, NOT 500. Answer (B)

EXAMPLE 931K aparticülarstore,thenum er o computerss01dtheweekofB1ackFridaywas 70,

ilY!nber of compuier$ sold •thefrevioyg week Was 920. 'Which •of the'following bést:ppproxtmatés th
rcenft Increase in computesalés the@revious week to Che week of OfåQkG1day?l

470 - 320
0.47
320

This time, the week of Black Friday is not the "original" basis for the percent change. We put the difference
over the previous week's number, 320. The answer is (D)

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CHAPTER 2 PERCENT

A few more examples involving percent:

EXAMPLÉ 10: Thenumber of students at aschool decreased fronq 2010 tox201T. Vthe Kumber of •E
students was k, whifh9t@iefdlowing expgesses+engmber of students enrolledjn
Of

cn.25Je V)1.5k

The answer is NOT 1.20k. Percent change is based off of the original value (from 2010) and not the new value.
Let x be the number of students in 2010,
.80x k

x 1.25k

Therefore, there were 25% more students in 2010 than in 2011. Answer (C) .

fah$;2ÖO/Q SOSS$

We don't know the number of 10th graders at the school so let's suppose that it's 100.

Red Sox fans 40% of 100 40

Celtics & Red Sox fans 20% of 40 8

The answer is then ¯

100

A common strategy in percent questions is to make up a number to represent the total, typically 100.

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CHAPTER EXERCISE: Answers for this chapter start on page 255.

A calculator is allowed on the following

questions. Questions 4-7 refer to the following
information.

The table below shows the number of box spring

If x is 50% larger than z, and y is 20% larger than and mattress units sold over four weeks at a
z, then x is what percent larger than y? bedding store.

A) 15% Week 1 2 3 4 Total

B) 20% Box Springs 42 34 167
C) 25% Mattresses 47 61 68 43 219
Total 85 103 121 77 386

Veronica has a bank account that earns m% Which week accounted for approximately 32%
interest compounded annually. If she opened the of all the box spring units sold?

account with $200, the expression $200 (x)t A) Week 1

represents the amount in the account after t
B) Week 2
years. Which of the following gives x in terms of
C) Week 3
D) Week 4

D) 1 +100m
Approximately what percentage of all units sold
came from week 2?
A) 15.8%
B) 26.7%
In a survey of 400 seniors, x percent said that
C) 31.3%
they plan on majoring in physics. One university
has used this data to estimate the number of D) 47.0%
physics majors it expects for its entering class of
3,300 students. If the university expects 66
physics majors, what is the value of x?

Mattresses accounted for approximately what

percentage of all units sold during week 1?
A) 22.0%
B) 32.5%

C) 44.7%
D) 55.3%

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CHAPTER 2 PERCENT

10

What was the approximate percent decrease in Joanne bought a doll at a 10 percent discount off
the number of mattresses sold from week 3 to the original price of $105.82. However, she had
week 4? to pay a sales tax of x% on the discounted price.
If the total amount she paid for the doll was
A) 37%
$100, what is the value of x?

D) 58%

The discount price of a book is 20% less than the

number of houses built in Town A
In 2010, the
retail price. James manages to purchase the book
was 25 percent greater than the number of
at 30% off the discount price at a special book
houses built in Town B. If 70 houses were built in
sale. What percent of the retail price did James
Town A during 2010, how many were built in
pay?
Town B?
A) 420/0

B) 480/0

D) 56%

12
Each day, Robert eats 40% of the pistachios left
in his jar at that time. At the end of the second Over a two week span, John ate 20 pounds of
day, 27 pistachios remain. How many pistachios chicken wings and 15 pounds of hot dogs. Kyle
were in the jar at the start of the first day? ate 20 percent more chicken wings and 40

A) 75 percent more hot dogs. Considering only

chicken wings and hot dogs, Kyle ate
B) 80
approximately x percent more food, by weight,
C) 85 than John. What is x (rounded to the nearest
D) 95 percent) ?

A) 25
B) 27

C) 29
D) 30

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13 16

Jane is playing a board game in which she must A small clothing store sells 3 different types of
collect as many cards as possible. On her first accessories: 20% are scarves, 60% are ties, and
turn, she loses 18 percent of her cards. On the the other 40 accessories are belts. If half of the
second turn, she increases her card count by 36 ties are replaced with scarves, how many scarves
percent. If her final card count after these two will the store have?
turns is n, which of the following represents her
starting card count in terms of n?

n
(1.18) (0.64)

B) (1.18) (0.64)n
n
C)
(1.36) (0.82) 17
D) (0.82) (1.36)n
Daniel has $1000 in a checking account and
$3000 in a savings account. The checking
account earns him 1 percent interest
compounded annually. The savings account
Due to deforestation, researchers expect the deer earns him 6 percent interest compounded
population to decline by 6 percent every year. If annually. Assuming he leaves both these
the current deer population is 12,000, what is the accounts alone, which of the following
approximate expected population size 10 years represents how much more interest Daniel will
from now? have earned from the savings account than from
the checking account after 5 years?
A) 4800
B) 6460
C) 7240
D) 7980 C) (3, 000(1.06) 5 3, 000)
1, 000)

D) (3, -3,000)
15
(1, 000(1.01) (5) - 1, 000)

Kyle bought a $2,000 government bond that

yields 6% in simple interest each year. Which of
the following equations gives the total amount
A, in dollars, Kyle will receive when he sells the Kristen opens a bank account that earns 4%
bond after t years? interest each year, compounded once every two
A) A— + .06)t years. Ifshe opened the account with k dollars,

B) A— + 0.060 which of the following expressions represents

the total amount in the account after t years?
C) A= + 0.06)t
2, ooo(l + 0.06)

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 Exponential vs. Linear
Growth
The population of ants doubling every month. A bank account earning 5 percent every year. These are
examples of exponential growth, which occurs when the amount at each stage is multiplied by a number
greater than 1. In the case of the ants, this number is 2. In the case of the bank account, it's 1.05. When
exponential growth happens, we can model it as a function that looks like

Y = ax
where y is the final amount after t time intervals, a is the initial amount, and x is the rate that we multiply by.
So if we started off with 100 ants, our equation would be

where t is the number of months that have gone by. And if our bank account started off with $200, our equation
would look like

where t is the number of years. You've seen this already in the previous chapter.

Graphs of exponential growth have the following shape:

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Notice how the graph creeps up slowly at first but then shoots up faster and faster over time. That's exponential
growth.
Exponential decay, however, is the opposite. Imagine a radioactive element that loses mass over time. It loses
a lot of its mass at first but then loses it more slowly over time.

Mass

Memorize the shape of these graphs for exponential growth and decay. The SAT will test you explicitly on
them.

The equation for exponential decay is the same as the equation for exponential growth:

Y — ax
The only difference is that the rate, x, is less than 1. So in the case of radioactive decay, the equation might look
like

where y is the final mass, 400 is the initial amount, and t is the number of years that have gone by.

Now compare exponential growth and decay to linear growth and decay. As you may already know, linear
growth can be modeled by a line with a positive slope. For example, if Ann has a piggybank with 50 dollars
already in it, and she adds 10 dollars every month, the total amount in the piggybank can be modeled by

A 10t +50

where A is the total amount, t is the number of months, and 50 (the y-intercept) is the initial amount.

Unlike exponential growth, linear growth doesn't have moments when it slows down or speeds up. Growth
is constant. It goes up by the same amount each time.

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CHAPTER 3 EXPONENTIAL VS. LINEAR GROWTH

The same holds for linear decay. Imagine Ann now takes 10 dollars every month out of her piggybank, which
initially contained 100 dollars. The final amount A would be

A = 50 10t

The decrease is at a constant rate, and the slope is negative.

Both exponential decay and linear decay are examples of a negative association between two things. As one
thing increases, the other thing decreases. For example, the number of absences over the semester and final
exam scores:
Final Exam Score

Number of Absences

When the data points are close to forming a smooth line or graph that shows the negative relationship, we can
say there is a strong negative association.

A positive association happens when one thing increases, the other thing also increases. We saw this with
exponential growth and linear growth. For example, the number of hours spent studying and final exam
scores:

Final Exam Score

Hours Studied

The graph above shows a positive association that is quite strong.

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 THE COLLEGE PANDA

CHAPTER EXERCISE: Answers for this chapter start on page 257.

A calculator should NOT be used on the

following questions.
The population of trees in a forest has been
decreasing by 6 percent every 4 years. The
population at the beginning of 2015 was
If the initial population of rats was 20 and grew estimated to be 14,000. If P represents the
to 25 after the second year, which of the population of trees t years after 2015, which of
following functions best models the population
the following equations gives the population of
of rats P with respect to the number of years t if
trees over time?
the population growth of rats is considered to be
exponential?
A) P— 14,
= 14, 000 + 0.94(4t)
A) P—5t+20
14, 000(0.94)

14,

D) +20

If was 100 and

the initial population of pandas
grew to 125 after the which of the
first year,

following functions best models the population

of pandas P with respect to the number of years
t if the population growth of pandas is

considered to be linear?

A) P — 25t + 100

— —I— 5t 100

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CHAPTER 3 EXPONENTIAL VS. LINEAR GROWTH

A calculator is allowed on the following Jamie owes Tina some money and decides to pay
her back in the following way. Tina receives 3
questions.
dollars the first day, 6 dollars the second day, 18
and 54 dollars the fourth
dollars the third day,
day. Which of the following best describes the
Which scatterplot shows the strongest positive relationship between time and the total amount

association between x and y? of money (cumulative) Tina has received from

Jamie over the course of these four days?

A) Increasing linear
B) Decreasing linear
C) Exponential growth
D) Exponential decay

Albert has a large book collection. He decides to

B) two of his used books for one new book
trade in
each month at a local bookstore. Which of the
following best describes the relationship
between time (in months) and the total number
of books in Albert's collection?

A) Increasing linear
B) Decreasing linear

C) Exponential growth
D) Exponential decay
C)

A scientist counts 80 cells in a petri dish and

finds that each one splits into two new cells
every hour. He uses the function A (t) cri to
calculate the total number of cells in the petri
dish after t hours. Which of the following
x
assigns the correct values to c and r?
D) A) —2
B) c — 80,r — 0.5

c — 80 r 1.5

D) c = 80,r —2

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 THE COLLEGE PANDA

Of the following scenarios, which one would

result in linear growth of the square footage of a
store?

A) The owner increases the square footage by

0.75% each year.
B) The owner increases the square footage by
5% each year.
C) The owner expands the store by 5% of the
original square footage each year.

D) The owner alternates between adding 200

square feet one year and 300 square feet the
next year.

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 Proportion
Imagine we have a triangle. We know that the area of a triangle is A —bh.

Now let's say we triple the height. What happens to the area?
Well, if we triple the height, the new height is 3h. The new area is then

1
Anew -b(3h) =3 -bh 3A01d
2

See what happened? The terms were rearranged so that we could clearly see the new area is three times the
old area. We put the "3" out in front of the old formula.
This technique is extremely important because it saves us time on tough proportion problems. We could've

made up numbers for the base and the height and calculated everything out, and while that's certainly a
strategy you should have in your toolbox, it would've taken much longer and left us more open to silly
mistakes.

Let's do a few more complicated examples.

EXAMPLE the radius :ofÄ grcle IS mereasyg by 250K By What percexydøes the areæofthe circle

Let the original area be Aold. If the original radius is r, then the new radius is 1.25r.

Anew — 1.5625(7tr2) — 1.5625A01d

We can see that the area increases by

The idea is to get a number in front of the old formula. In the previous example, that number turned out
tobe 1.5625. Also note that the 1.25r was wrapped in parentheses so that the whole thing gets squared. It

would've been incorrect to have Anew because we wouldn't be squaring the new radius.

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 THE COLLEGE PANDA

EXAMPLE 2', The:length Of arectangle i$ increased by 200/0.•ThéNY1d$h is decreased >y200/o. Which Of

follOVing accurately:descfibes the •iO the area ofthe rectangle?

Originally, A lw. Now,

new — (1.201) (0.80w) 0.961w 0.96A01d

The area has decreased by 4%. Answer (C) . Most students think the answer is (D). It's not.

EXAMPLE3:

two 'particles Can pe deterrnined pygthe ormula above, whickv$ iS thet$

The«force of attraction between
between then, iS åistanee between of'thötwo particle
the distance bewveentvvo Oharged particles iS doubled) the fesUIfing forct of'ttraétiOn is what fraction Of

theoriymdforce?
C)

(1h2 (9mq2 1 9q1q2

new Fold
(2r)2

Answer (B) . Notice how we do not let constants like the "9" in the formula affect the result. In getting

a number out front, students often make the mistake of mixing that number up with numbers that were
originally in the formula.

EXAMPLE Q of a The length ofeach.sYde.mUsthavebeen Increase by

whaepergent?

Fj12% C)Å3Yo,

Now we have to solve backwards. Keep in mind that the volume of a cube is V s3 where s is the length of
each side. Even though this problem is a little different, we can still apply the same process as before: increase
each side by some factor and rearrange the terms to extract a number. Only this time, we have to use x.

new — (xs)3

new X
33
S x3V01d

Notice how we were still able to extract something out in front, x3. That x3 must be equal to 3 if the new
volume is to be triple the old volume:

x - 1.44

Each side must have been increased by approximately 44%. Answer (D)

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CHAPTER 4 PROPORTION

CHAPTER EXERCISE: Answers for this chapter start on page 258.

A calculator is allowed on the following

questions.
bl c

power P is related to the voltage V and

Electric
resistanceR by the formula above. If the voltage
were halved, how would the electric power be b2
affected?
The area of the trapezoid above can be found
A) The electric power would be 4 times greater.
B) The electric power would be 2 times greater.
using the formula — (bl + b2)h. If lengths BC and
C) The electric power would be halved. AD are halved and the height is doubled, how
would the area of the trapezoid change?
D) The electric power would be a quarter of
what it was. A) The area would be increased by 50 percent.
B) The area would stay the same.

C) The area would be decreased by 25 percent.

D) The area would be decreased by 50 percent.
Julie has a square fence that encloses her garden.
She decides to expand her garden by making
each side of the fence 10 percent longer. After
this expansion, the area of Julie's garden will
have increased by what percent? Calvin has a sphere that is four times bigger
than the one Kevin has in terms of volume. The
radius of Calvin's sphere is how many times
greater in length than the radius of Kevin's
C) 22% sphere (rounded to the nearest hundredth)?

B) 1.59

C) 1.67
D) 2.00
A right circular cone has a base radius of r and a
height of h. If the radius is decreased by 20
percent and the height is increased by 10
percent, which of the following is the resulting
percent change in the volume of the cone?

A) 100/0

B) 12% decrease
C) 18.4% decrease
D) 29.6% decrease

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 THE COLLEGE PANDA

Astronomers see two equally bright stars, Star A

and Star B, in the night sky, but the luminosity of
450 Star A is one-ninth the luminosity of Star B. The
S
distance of Star A from Earth is what fraction of
the distance of Star B from Earth?
1

450 27
1
s
B)
9
In the triangle above, the lengths of the sides 1
relate to one another as shown. If a new triangle C)
3
is created by decreasing s such that the area of 2
the new triangle is 64 percent of the original area, 3
s must have been decreased by what percent?

B) 200/0

C) 25%

Questions 7-8 refer to the following

information.

L 47td2b

The total amount of energy emitted by a star each

second is called its luminosity L, which is related to
d, its distance (meters) away from Earth, and b, its

brightness measured in watts per square meter, by

the formula above.

If one star is three times as far away from Earth

as another, and twice as bright, its luminosity is

how many times greater than that of the other

Star?

C) 16
D) 18

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 Rates
I've found rate problems to be pretty polarizing—some students just "get" them intuitively, others get completely
lost. Most of the rate problems on the SAT will be pretty straightforward, but for the ones that aren't, I highly
recommend using conversion factors to setup the solution (if you've gone through chemistry, you should know
what I'm talking about). Conversion factors are a fool-proof way to approach a lot of these problems, but they
can be slow-going for stronger problem solvers. I'll be covering both the straightforward, intuitive approaches
and the conversion factor approach throughout the examples in this chapter.

EXAMPLE& bicyclesperhoumHownanyhourswou14ittake
thejnanUfaemrefto

Easy enough. We divide the total by the rate to get 320 + 20 — hours.

EXAMPLE 23Aorockethas 360 offuelleftaite '

of flight* It yurns•ngallonsof a constant. What is thesvalue of n?

Here, we are figuring out the rate. In 6 —2 4 hours of flight, the rocket burned 360 100 = 260 gallons of
260
fuel. Therefore, the rocket burns gallons of fuel every hour.
4

EXAMPLE3:t attwe supermatket Costs 20 tila€

thesupemarke!fi oranges„hOwnanybOxeswillthesupermarkéEbeåble to
co*ieiy fill?

If each orange is 20 cents, then a dollar would be enough for 5 oranges. Five hundred dollars would be enough
for 500 x 5 2500 oranges, which would fill 2500 + 6 = 416.67 boxes. Given that the question asks for full

boxes, the answer is 416

The examples above were quite straightforward and didn't really call for writing out full conversion factors,
but what if we wanted to use conversion factors for Example 3? What would've the solution looked like?

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 THE COLLEGE PANDA

100 cerrtS 1 grange 1 box

500 x 416.67 boxes
1 doffäf 20 centS 6 granges

The rest of the examples in this chapter are done with conversion factors to teach you how they're used, even
though there may be more "casual" solutions.

EXAMPLE 4: I-mile iruxninute and 15 seconds. At this rate,how manymiles can the car
travel in

In most rate problems, you'll start with what the question is asking for. We need to convert that 1 hour to a
distance that the car travels. The car's rate is 1 mile every 75 seconds.

60 mintiteS 60 second' 1 mile 60 >< 60 miles

miles
1 hotlf 1 m-irrüt€ 75 seeondé 75

The units should cancel as you go along. If the units are canceling, chances are we're doing things right. Notice
that the "miles" unit at the end is the unit we wanted to end up with. This is another sign that we've done

things right.

'EXAMPLE G: Torn hes3(j åt an averagefaG of 50 miles per hour: If Leona drives at an åverage
xate»014dmi1es pernour,howmanymoreminutgswi111tÅk her to Såme

We have to figure out how long it takes Tom to drive 30 miles:

1 hour 60 minutes
30 miles x 36 minutes
50 miles 1 hour

Leona will take

1 hour 60 minutes
30 miles >< 45 minutes
40 miles 1 hour

so,
45 - 36 9 minutes

MPLE6$TOVrepare for print anumberofbookletswith pages perbookletvlf

everybpagescostccentéto priHCand he spentat&fai of nmanyboo eCsdid Chuprmå
terms of and d?

B)
Good bye p

100 centS 5 pages 1 booklet 500d

booklets
1 deHäf c cerrtS p pages cp

The answer is (C)

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CHAPTER 5 RATES

CHAPTER EXERCISE: Answers for this chapter start on page 260.

A calculator should NOT be used on the

following questions.
An electronics company sells computer monitors
and releases a new model every year. With each
new model, the company increases the screen
A carpenter lays x bricks per hour for y hours size by a constant amount. In 2005, the screen
and then lays — bricks for 2y more hours. In size was 15.5 inches. In 2011, the screen size was
18.5 inches. Which of the following best
terms of x and y, how many bricks did he lay in describes how the screen size changed between
total?
2005 and 2011?
A) 2xy
A) The company increases the screen size by
0.5 inch every year.
B) The company increases the screen size by 1

C) 5xy inches every year.

C) The company increases the screen size by 2

inches every year.

D) The company increases the screen size by 3

inches every year.

day. If Tim were to meet this requirement by

only eating a certain protein bar that contains 30
As a submarine descends into the deep ocean,
grams of protein, how many protein bars would the pressureit must withstand increases. At an
he have to buy to last a week?
—700 meters, the pressure is 50 atm
altitude of
(atmospheres), and at an altitude of —900
meters, the pressure is 70 atm. For every 10
meters the submarine descends, the pressure it

faces increases by n, where n is a constant. What

is the value of n?

A) 0.1

D) 10

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 THE COLLEGE PANDA

An empty pool can be filled in 5 hours if water is During a race on a circular race track, a racecar
pumped in at 300 gallons an hour. How many burns fuel at a constant rate. After lap 4, the
hours would it take to fill the pool if water is racecar has 22 gallons left in its tank. After lap 7,
pumped in at 500 gallons an hour? the racecar has 18 gallons left in its tank.
Assuming the racecar does not refuel, after
which lap will the racecar have 6 gallons left in
its tank?

A) Lap 13
B) Lap 15
C) Lap 16
D) Lap 19

If a apples cost d dollars, which of the following

expressions gives the cost of 20 apples, in
dollars?
By 1:00 PM, a total of 40 boxes had been
20a unloaded from a delivery truck. By 3:30PM, a
total of 65 boxes had been unloaded from the

20d same truck. If boxes are unloaded from the truck

at a constant rate, what is the total number of
boxes that will have been unloaded from the
20d truck by 7:OOPM?

20
ad

At a school, there are a grade levels with b

students in each grade. If the school buys n
10
stickers tobe distributed equally among the
students, which of the following gives the
number of stickers each student receives?
Amy buys d dollars worth of groceries each
week and spends a fourth of those dollars on
ab fruit. In terms of d, how many weeks will it take
Amy to spend a total of $100 just on fruit?
an
400
b
d
bn
25
a B)
n
D) d
ab
25
d
400

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CHAPTER 5 RATES

11 14

An internet service provider charges a one time A cupcake stole employs bakers to make boxes
setup fee of $100 and $50 each month for service. of cupcakes. Each box contains x cupcakes and
If c customers join at the same time and are on each baker is expected to produce y cupcakes
the service for m months, which of the following each day. Which of the following expressions
expressions represents the total amount, in gives the number of boxes needed for all the
dollars, the provider has charged these cupcakes produced by 3x bakers working for 4
customers? days?

A) 100c + 50m A) 12x2y

B) 100c + 50cm 3y
C) 150cm 4

D) 100m + 50cm 12X2

C)

12 D)

A manufacturing plant increases the

temperature of a chemical compound by d
degrees Celsius every m minutes. If the
At a math team competition, there are m schools
compound has an initial temperature of t
with n students from each school. The host
degrees Celsius, which of the following
school wants to order enough pizza such that
expressions gives its temperature after x
there are 2 slices for each student. If there are 8
minutes, in degrees Celsius?
slices in one pizza, which of the following gives
the number of pizzas the host school must
d order?

md + t mn
B) 8
x
mn
4
mx
C)
8
m
D) 2mn

13

At a shop for tourists, the price of one souvenir

is a dollars. Each additional souvenir purchased

after the first is discounted by 40 percent. If

James buys n souvenirs, where n > 1, which of
the following represents the total cost of the
souvenirs?

D) 0.6m

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 THE COLLEGE PANDA

16 19

Idina can type 90 words in 2.5 minutes. How

many words can she type in 12 minutes?
The expression above gives the population of
leopards after five years during which an initial
population of P leopards grew by r percent each
year. Which of the following expressions gives
the percent increase in the leopard population
over these five years?

20

—1 A painter can cover a circular region with a

100 x 100
B) radius of 3 feet with paint in 2 minutes. At this
100 rate, how many minutes will it take the painter
to cover a circular region with a radius of 6 feet
1 x 100 with paint?
{öö)5

x 100
100

A calculator is allowed on the following

questions. 21

A "slow" clock falls behind at the same rate

Henry drives 150 miles 30 miles per hour and
at every hour. It is set to the correct time at 4:00
then another 200 miles at 50 miles per hour. AM. When the clock shows 5:00 AM the same
day, the correct time is 5:08 AM. When the clock
What was his average speed, in miles per hour,
for the entire journey, to the nearest hundredth? shows 10:30 AM that day, what is the correct
time?
A) 38.89
B) 4000
A) 11:02 AM
B) 11:18 AM
C) 42.33
C) 11:22 AM
D) 43.58
D) 12:18 PM
18
22

A rolling ball covers a distance of 2400 feet in 4

minutes. What is the ball's average speed, in A salesman at a tea company makes a $15
inches per second? (12 inches = 1 foot) commission on every $100 worth of products
that he sells. If a jar of tea leaves is $20, how
many jars would he have to sell to make $180 in
commission?

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CHAPTER 5 RATES

23 26

A train covers 32 kilometers in 14.5 minutes. If it An 8 inch by 10 inch piece of cardboard costs
continues to travel at the same rate, which of the $2.00. If the cost of a piece of cardboard is

following is closest to the distance it will travel proportional to what is the cost of a
its area,
in 2 hours? piece of cardboard that is 16 inches by 20 inches?

A) 54 kilometers A) $4.00
B) 265 kilometers B) $8.00

C) 364 kilometers C) $12.00

D) 928 kilometers D) $16.00

24 27

One liter is equivalent to approximately 33.8 Margaret can buy 4 jars of honey for 9 dollars,
ounces. Mark has plastic cups that can each hold and she can sell 3 jars of honey for 15 dollars.
12 ounces of liquid. At most, how many of these How many jars of honey would she have to buy
plastic cups could a two liter bottle of soda fill? and then sell to make a total profit of 132
dollars?

25

28
Brett currentlyspends $160 each month on gas.
His current car is able to travel 30 miles per
In one hour, Jason can install at least 6 windows
gallon of gas. He decides to switch his current
but no more than 8 windows. Which of the
car for a new car that is able to travel 40 miles
following could be a possible amount of time, in
per gallon of gas. Assuming the price of gas
hours, that Jason takes to install 100 windows in
stays the same, how much will he spend on gas
a home?
each month with the new car?
A) 12
A) $100
B) 16
B) $120
C) 17
C) $130
D) 18
D) $140

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29

1 fluid ounce — 29.6 milliliters

1 cup 16 fluid ounces

A chemistry teacher is planning to run a class

experiment in which each student must measure
out 100 milliliters of vinegar in a graduated
The class is limited to using 6 cups of
cylinder.
Given the information above, what is
vinegar.
the maximum number of students who will be
able to participate in this experiment?

30

Yoona runs at a steady rate of 1 yard per second.

Jessica runs 4 times as fast. If Jessica gives Yoona
a head start of 30 yards in a race, how many
yards must Jessica run to catch up to Yoona?

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 Expressions
3m — k
Algebraic expressions are just combinations of numbers and variables. Both x2 + y and are examples
2
of expressions. In this chapter, we'll cover some fundamental techniques that will allow you to deal with
questions involving expressions quickly and effectively.

1. Combining Like Terms

When combining like terms, the most important mistake to avoid is putting terms together that look like they
can go together but can't. For example, you cannot combine b2 + b to make b3, nor can you combine a + ab to
make 2ab. To add or subtract, the variables have to completely match.

EXÄMPtÉn:

Whichof thefOllowingis equivalent totheexpre ionabove?

P)3ai- lla±bi

2(2a2 - 3a2b2 4b ) — ( a2 + 5a2b2 10b2) 4a2 6a2b2 — 8b2 - 5a2b2 + 10b2

— 3a 2 — Ila2b2 2172

Answer (C) .

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 THE COLLEGE PANDA

2. Expansion and Factoring

EXAMPLE 2:
•-34j(2x4Ä3)

Which of the followingis equivalent to the expression above?

.24 - B)

Some people like to expand using a method called FOIL (first, outer, inner, last), If you haven't heard of it,

that's totally fine. After all, it's the same thing as distributing each term. First, we distribute the 02."

+3) (21 - + 3)

Notice that it applies to just one Of the two factors. Either one is fine, but NOT both.

(2x — 4x2 + 61- 16x -24

- lox -24

Answer (A)

Now when it comes to factoring and expansion. there are several key formulas you should know:

• (a + + 2nb + b2
• (a — 2ab b2

Memorize these forwards and backwards. They show up very often.

EXAMPLE Whiehof the following is equivalent to 9y2y

— pyj

Part of what makes for a top SAT score is pattern recognition. Once you've done enough practice, you should
be able to recognize the question above as a difference Of two squares, a variation Of the — b2 forrnula.
SAT will rarely test you on those formulas in a straightforward way Be on the lookout for variations that
match the pattern. With more practice, you'll get better and better at noticing them.
Using the formula — b? (a + b) (a — b), we can see that 2.v2 and b 3y, 'Therefore,

41-4 — 9y2 (2r2 — 3y)

Answer (D)

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CHAPTER 6 EXPRESSIONS

EXAMPLE
16x4 •8xXy2

WhicKof thefonowmg expresston

ZÅj(4x2 4y2)å yr.

Using the formula (a — 2

2ab + b2 (in reverse), we can see that a = 4x2 and b — y2. Therefore,

16x4 — 8x2y2 + y4 = (4x2 — y2)2

This is not in the answer choices. We have to take it one step further and apply the a2 — b2 formula to the
expression inside the parentheses.

Answer (C) .

3. Combining Fractions

When you're adding simple fractions,

1 1

3 4

the first step is to find the least common multiple of the denominators. We do this so that we can get a common
denominator. In a lot of cases, it's just the product of the denominators, as it is here, 3 x 4 — 12.
1 1 1413 4 3 7
-4 •j 12 12 12

Now when we're adding fractions with expressions in the denominator, the idea is the same.

NEXAMPtÉ5•,

eqUWalent Vo the
3x 2
B)

The common denominator is just the product of the two denominators: (x + 2) (x — 2). So now we multiply
the top and bottom of each fraction by the factor they don't have:

1 2 1 2

Answer (B) .

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 THE COLLEGE PANDA

4. Flipping (Dividing) Fractions

1
1
2
What's the difference between and
3 2
3

2
The difference is where the longer fraction line is. The first is divided by 3. The second is 1 divided by
3
They're not the same.
1

2 1 1 1 1

3 2 2 3 6

1 2 3 3
2 3 2 2
3

The shortcut is to flip the fraction that is in the denominator. So

a ac
b b

If the fraction is in the numerator, then the following occurs:

a
a
c bc

•EX'MPLE t: If x which-of the followingis eqtli&ayn/

VI IYCx 1

First, combine the two fractions on the bottom with the common denominator (x — 1) (x + 1).
1 1

Next, substitute this back in and flip it.

x x(x
2

Answer (D) .

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CHAPTER 6 EXPRESSIONS

5. Splitting fractions

30+c
•EXAMPLE'; Which Of the following is equivalent to

B) C) 5+c— D)5+

We can split the fraction into two:

30 c 30 c c

6 6 6 6

The answer is u. This is just the reverse of adding fractions.

Note that while you can split up the numerators of fractions, you cannot do so with denominators. So,

3
In fact, you cannot break up a fraction like any further.

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 THE COLLEGE PANDA

CHAPTER EXERCISE: Answers for this chapter start on page 263.

A calculator should NOT be used on the

following questions.
Which of the following is equivalent to
12x

Which of the following is equivalent to 3x

6x2y + 6xy2? 4 + 2x
B)
A) 6xy(x + y) 3x

B) 12xy(x +y) C)
3x
C) 6x2y2(y + x)
D) 1
D) 12x3y3

If a > 0, then — + — is equivalent to which of the

Which of the following is equivalent to 3x4 — 3?

following?
B) 3(x2 -1)2

B)

4
D)
Which of the following is equivalent to the
expression shown above?

Which of the following is equivalent to

(x2 + y) (y + z)?
A) x2Z —k Y2 YZ

B) x2y + x2z + Y2 + yz
C) X2y+y2+x2z
2
D) x2 —I— x2Z Y2 —I— Yz xy x
Which of the following is equivalent to
xy — Y2

x
x

x
D)

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CHAPTER 6 EXPRESSIONS

If x > 1, which of the following is equivalent to A calculator is allowed on the following

1
questions.

2 3 10

3x3 8x — 4x
6
6 7x2 - lix -7
B)
Which of the following is the sum of the two
6 polynomials above?
C)
5x +7 + x2 — 15x - 7
A) 3x3
1
D) B) 3x3 + 15x2 - 15x - 7
42
C) 10x5 - 7x - 7

D) 15x4 + 3x3 — 15x2 - 7

1 11
x
1
2 (5a + 3 vna) (2a +
x
Which of the following is equivalent to the
The expression above is equivalent to which of expression above?
the following?
A) — 2av/å

B)

C)
12

If y 0, what is the value of

8 (3y)2

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 Manipulating &
Solving Equations
On the SAT, there is a huge emphasis on equations. To get these types of questions right, you must learn how
to isolate the variables and expressions you want. First, we'll cover several useful techniques in dealing with
equations that you may already be familiar with.

1. Don't forget to combine like terms

You should be ruthless in finding like terms and combining them. Doing so will simplify things and allow you
to figure out the next step.

EXAMPLEI:

The same four variables are on both sides of the equation, a, b, c and d. That should tell you to distribute on the
left side first and then combine like terms. Sounds simple but you won't believe how many students forget to

do this, especially in the middle of a more complex problem.

The b, c, and d variables cancel quite nicely.

2a 2 317

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CHAPTER 7 MANIPULATING & SOLVING EQUATIONS

2. Square and square root correctly

When squaring equations to remove a square root, the most important thing to remember is that you're not
squaring individual elements you're squaring the entire side.

EXAMPLE 2:

31f (j and b tyeeqyatiorvgboveis *followmgi"'

b)åb * al%+b2 9iab -a2

The square root in the problem should scream to you that the equation should be squared. Most students know
the square root should be eliminated, but here's the common mistake they make:

ab a
2 — b2

They square each individual element. However, this is WRONG. When modifying equations, you must apply
any given operation to the entire SIDE, like so:

If it helps, wrap each side in parentheses before applying the operation. By the way, the same holds true for
all other operations, including multiplication and division. When you multiply or divide both sides of an
equation, what you're
actually doing is wrapping each side in parentheses, but because of the distributive
property, ithappens that multiplying or dividing each individual element gets you the same result. For
just so
example, if we had the equation

and we wanted to multiply both sides by 3, what we're actually doing is

which turns out to be the same as

Anyway, back to the problem

( ab)2 (a —
2
ab a — 2ab + b2
3ab

The answer is D
Another common mistake is squaring each side before the square root is isolated on one side. For example,

Don't square each side until you've moved the "3" on the left to the right:

And now we can square both sides and go from there.

Now, when it comes to taking the square root of an equation, most students forget the plus or minus

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 THE COLLEGE PANDA

Always remember that an equation such as x2 25 has two solutions:

v/iS

However, this only applies when you're taking the square root to solve an equation. By definition, square
roots always refer to the positive root. So, — 3, NOT And — —3 is not possible (except when
working with non-real numbers, which we'll look at in a future chapter). The plus or minus is only necessary
when the square root is used as a tool to solve an equation. That way, we get all the possible solutions to the
equation.

EXAMPLE3: If (E +3)2 121, what isothe,sum ofthe two possible values of x?

(x +3)2 — 121

— 121

¯ ill
11

So x could be either 8 or —14. The sum of those two possibilities

3. Cross-multiply when fractions are set equal to each other

Whenever a fraction is equal to another fraction,

a c

you can cross-multiply: ad — bc.

E MPLE4tif x what is the value ofx?

4 10
—x
5 3

12x 50

25
6

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CHAPTER 7 MANIPULATING & SOLVING EQUATIONS

EXAMPLE5i1f — what istheovalue of X?

5 3
—o
5 3

5x + 10 3x —6
—16

-8

4. Factoring should be in your toolbox

Some equations have variables that are tougher to isolate. For a lot of these equations, you will have to do
some shifting around to factor out the variable you want.

'filch of the followin of band"c?

a
b

3ab + bc
bc =a 3ab

bc - 3b)
bc
a
1-3b

See what we did? We expanded everything out and put every term containing a on the right side. Then we
were able to factor out a and isolate it. The answer is (A)

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 THE COLLEGE PANDA

GXAMPLE 7:

—What is one possible real valüe of x for which the equation above is true?

x4 + 3x3 + x + 3
x3(x +3) + (x +3) —
0

x —3 or — 1

Once we factored out x3 from the first two terms, further factoring was possible with the (x + 3) term. How
would you know to do this? Experience.

5. Treat complicated expressions as one unit

EXAMPLE 8:

+ xæ:+r

ich thefollowing gives m. in terms

Don't let the big and complicated expressions freak you out. Treat these complicated expressions as one unit
or variable, like so:

Multiply both sides by m.

mA
c
Divide both sides by A.

AC

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CHAPTER 7 MANIPULATING & SOLVING EQUATIONS

Finally, plug the original expressions back in.

x
1

Answer (D)

99
+ 24 — 0

:rfxs for what real value Of the equation abovetrue?

Treat (x + 1) as one unit and call it A.

-24=0
- 24
—o
—o

-9 or 2

Because the question stipulates that x > 0, the answer is 2 .

6. Be comfortable solving for expressions, rather than any one variable

EXAMP If Bx 4-9y tge 3y?e

Get in the habit of looking for what you want before you solve for anything specific. Is there any way to get
the answer without solving for x and y?

Yes! Dividing both sides of the given equation by 3 gives x + 3y=@

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 THE COLLEGE PANDA

EXAMPLE 11J1f x 3, whatis the ValueOfrY2

2
B) D)

Here, we have no choice but to solve for the expression. We're given x over y but we want y over x. We can
flip the given equation to get
1

Then we can divide both sides by 2 to obtain the — we're looking for.

1 1

2-3 6

The answer is (A)

7. Guess and check when you're out of options

When all else fails and you don't have any answer choices or a calculator to work from, it never hurts to guess
and check small numbers.

xpe 4) 4x

If XIS integer, vhati$songpossiblesoluuon O théeqtå@tion

If you have to do a question this complicated without a calculator or any answer choices, you know it has to
be solvable through basic guess and check. There's simply no other way.

It would be silly to show you every step of guess and check on this page. Just remember to start with numbers
like 0, 1, 2, and —1. In this case, the answer is 2 .

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CHAPTER 7 MANIPULATING & SOLVING EQUATIONS

EXERCISE 1: Isolate the variable in bold. Answers for this chapter start on page 264.

1. 7tr2
2
22. If t —ax, find ax in terms of t.
3
2. C 27tr
23. If 3x + 6y — 7z, find x + 2y in terrns of z.
3. A —bh 24. If x 5 2b, find 2x + 10 in terms of b.

V Iwh
25. If a, find 4t in terms of a.

V nr2h
2
26. If find p
V 7tr2h 3'

2
7. c a2 + b2 1
27. If find t in terms of r.
2'
V— s3

28. If XY z, then find x2y in terms of z.

9. S — 27trh + 27tr2
a c 4x+1
10. 29. If 2 p(x5 — x4), what is p in terms of x?
d x3 — x

a c 1
11. 30. lf2X r3 m(x2 + 1) 2 , what is m in
l) x
terms of x?

1
31. If — x3 , what is n in terms of x?
13. m 512 —3 nx

32. 5(c + 1)3, what is a in terms of

14. m b and c?

7x2 3
15. u2 + 2as 33. If k(x2 4) -l- ky , what is k in terms of
2
a x x and y?
16.

34. If ax 3a x 3 b, what is x in terms of a

and b?
17. t — 27T

18.

19. If X find X in terms of Y and Z.

20. If x(y + 2) y, find y in terms of x.

21. If find a in terms of b and c.

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 THE COLLEGE PANDA

CHAPTER EXERCISE: Answers for this chapter start on page 264.

A calculator should NOT be used on the

following questions.
4 8
If m, what is the value of m?
9 3

1
—2, then (a + 6
2
3
5
C)

For what value of n is (n —

—8, what is the value of (x + 2)3?
A) -1
B) 1

D) 125

b 4 x
If 1, what is the value of b — ac? If where k —2, what is k in terms
a k+2 3'
of x?
A) —3
12
x
12 + 2X
D) It cannot be determined from the B)
x
information given.
x
C)
12 + 2x

If 3x -8 —23, what is the value of 6x

A) —5
2
21 If (x -3) 36 and x < 0, what is the value of
x2?
C) -30
¯ 37

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CHAPTER 7 MANIPULATING & SOLVING EQUATIONS

13

x2(x4 — 9) 8x4
l)
Ifx > 0, for what real value of x is the equation
The formula above gives the future value f of an above true?
annuity based on the monthly payment p, the
interest rate i, and the number of months n.

Which of the following gives p in terms of f, i,

and n?

1
B) 14

C)
If 6 and x > 0, what is the value of
—1 3
X?

m n
If 2, what is the value of
2m
1
15
8
1

4 20 — _ 10

1
Ifx > 0, for what value of x is the equation
2
above true?

11

If x < 0 and x2 — 12 4, what is the value of x?

A) — 16
B) —8

12

21, then what is the value of x2 + 3?

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 THE COLLEGE PANDA

A calculator is allowed on the following If 3 (x 2y) — 3z 0, which of the following

expresses x in terms of y and z?
questions.

16
3

If 8 + 5x is twice x — 5, what is the value of x?

D) 6y+3z
7
20
2
xy2+ x —1 0

If the equation above is true for all real values of

17
y, what must the value of x be?
x x + 12
If , what is the value of —?
6 42 x
1

18

Doctors use Cowling's rule, shown above, to

determine the right dosage d, in milligrams, of
medication for a child based on the adult dosage
a,in milligrams, and the child's age c, in years.
Ben is a patient who is in need of a certain
medication. If a doctor uses Cowling's rule to
prescribeBen a dosage that is half the adult
dosage, what is Ben's age, in years?

C) 11
D) 13

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CHAPTER 7 MANIPULATING & SOLVING EQUATIONS

Questions 21-22 refer to the following

information.
If the masses of both object 1 and object 2 were
doubled, how would the acceleration of the two
objects be affected?
1
A) The acceleration would stay the same.
B) The acceleration would be halved.
C) The acceleration would be doubled.
D) The acceleration would be quadrupled
(multipled by a factor of 4).

2 A

In the figure above, two objects are connected by a

string which is threaded through a pulley. Using its

weight, object 2 moves object 1 along a flat surface.

The acceleration a of the two objects can be

determined by the following formula

Tt12g — YTt11g

nil + 1112

where nil and Tt12 are the masses of object 1 and

object 2, respectively, in kilograms, g is the
acceleration due to Earth's gravity measured in
and ju is a constant known as the coefficient of
sec
friction.

21

Which of the following expresses in terms of

the other variables?

a(rnl + Tt12)
2
nil

a(nll + Tt12)
nt2g nilg

rt12g a(nll + 1112)

mig
a(ml + 1712) — nt2g
nilg

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 THE COLLEGE PANDA

Questions 23-24 refer to the following

information.

The value V of a car depreciates over t years

according to the formula above, where P is the
original price and r is the annual rate of

depreciation.

23

Which of the following expresses r in terms of

v, P, and t?

-1

24

If a car depreciates to a value equal to half its

original price after 5 years, then which of the

following is closest to the car's annual rate of
depreciation?

A) 0.13

B) 0.15

C) 0.16
D) 0.2

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More Equation Solving
Strategies
In this chapter, we'll touch on two equation solving strategies that are necessary for some of the tougher
questions.

1. Matching coefficients

It's hard to see anything meaningful right away on both sides of the equation. So let's expand the left side first
and sec if that takes us anywhere.

2
x 2ax a

So now we have
— x2 + 8x + b
We can match up the coefficients.
x2 + 2ax + a 2
So,

2a—8
2
a b

Solving the equations, a 4 and b

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2. Clearing denominators

When you solve an equation like —x + —x 10, a likely first step is to get rid of the fractions, which are harder

to work with. How do we do that? By multiplying both sides by 6. But where did that 6 come from? 2 times
3. Sothis is what you're actually doing when you multiply by 6:

1
-x. (2-3) + -x. (2-3) — 10 • (2-3)
2

1
(2.3) + -x. (2.3) = • (2 • 3)

3x + 2x 60

We got rid of the fractions by clearing the denominators. Here's the takeaway: we can do the same thing even
when there are variables in the denominators.

EXAMPLE 2:
—2
Ifx iSa$OIutiOn 0, is the Value Of x?

In the same way we multiplied by 2 3 before, we can multiply by x(x + 2)

• here.

3 5
x
3 5
— 2x(x + 2)
2x2 + 4x
— 2x2 + 4x

0
0

x 3 or x —1 but because x > 0, x

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CHAPTER 8 MORE EQUATION SOLVING STRATEGIES

Here's one final example that showcases both of the strategies in this chapter.

3x 01±+5

In the equation above—r and a IS a constant. What

A) —6 02

Let's clear the denominators by multiplying both sides by (x + 1) (ax + 2):

5 —6x2 5

3x 5 —6x2 I IX 5
x

3x(ax + 2) + 1) — —6x2 + 11 x + 5
—6x2 IIX 5

Comparing the coefficients of the x2 term on either side, 3a 6. Therefore, a —2. Answer (B) .

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CHAPTER EXERCISE: Answers for this chapter start on page 269.

A calculator should NOT be used on the

following questions.
If n < 0 and 4x2 + mx + 9 (2x + n)2, what is
the value of m + n?
If (2x + 3) (ax —5) — 12x2 + bx — 15 for all A) -15
values of x, what is the value of b?

D) 12

C) 10
D) 12

1 1
If — — , what is x in terms of p and y?

x2 + 99/2 + 42, what is the value of

x2y2? B)

py
C)
p¯Y
py
D)

If + 5 for all values of b, what is the

b In the equation above, a and b are integers. If the
value of a? equation is true for all values of x, what are the
two possible values of c?
A) 8 and 12
B) 14 and 21
C) 15 and 18
D) 17and23

1 1
If 1, what is the value of x?

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CHAPTER 8 MORE EQUATION SOLVING STRATEGIES

11

12x2 —I— 4-23 41 What is one possible solution to the equation

— 6x 18 +
12 2

In the equation above, m is a constant and X 4-2

What is the value of m?

A) -42
B) —36
C) -30
D) 42
12

4 2 35

(x 3 kx2 — 3) — 2) ¯ 18x2
Ifx > 1, what is the solution to the equation
above?
In the equation above, k is a constant. If the
equation is true for all values of x, what is the
value of k?

10

Ifn > 0, for what value of n is the equation

above true?

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 Systems of Equations
A system of equations refers to 2 or more equations that deal with the same set of variables.
—5x + y —
—12

There are two main ways of solving systems of 2 equations: substitution and elimination.

Substitution
Substitution is all about isolating one variable, either x or y, in the fastest way possible.
Taking the example above, we can see that it's easiest to isolate y in the first equation because it has no
coefficient. Adding 5x to both sides, we get
y—5x 7

We can then substitute the y in the second equation with 5x 7 and solve from there.

—3x - 2(5x - 7) — —12

— 3X IOX + 14 -12
13x — 26

Substituting x = 2 back into y = 5x 7, y 5(2) —7 3.

The solution is x 2, y 3, which can be denoted as (2, 3).

Elimination
Elimination is about getting the same coefficients for one variable across the two equations so that you can add
or subtract the equations, thereby eliminating that variable.

Using the same example, we can multiply the first equation by 2 so that the y's have the same coefficient (we

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CHAPTER 9 SYSTEMS OF EQUATIONS

don't worry about the sign because we can add or subtract the equations).

-lox + 2y — 14
-12

To eliminate y, we add the equations.

—lox + 2y — 14
-12

—13x —26

Now, we can see that x 2. This result can be used in either of the original equations to solve for y. We'll pick
the first equation.

-10(2) + 2y — 14
-20 + 2y — 14
6

And finally, we get the same solution as we got using substitution: x 2, y 3.

When solving systems of equations, you can use either method, but one of them will typically be faster. If
you see a variable with no coefficient, like in —5x + y —7 above, substitution is likely the best route. If
you see matching coefficients or you see that it's easy to get matching coefficients, elimination is likely the best
route. The example above was simple enough for both methods to work well (though substitution was slightly
faster). In these cases, it comes down to your personal preference.

No solutions
A system of equations has no solutions when the same equation is set to a different constant:

—4

The equations above contradict each other. There is no x and y that will make both of them true at the same
time. The system has no solution. Note that

6x + 4y -8

also has no solution. Why? Because the second equation can be divided by 2 to get the original equation.

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 THE COLLEGE PANDA

ax — 12y — 15

"if the system of solution, 'hat IS the value of a?

We must get the coefficients to match so that we can compare the two equations. To do that, we multiply the
second equation by —4:

—ax 12y — 15

-16x 12y 8

See how the —12's match now? Now let's compare. then we get our two contradicting equations
with no solution. One is set to 15 and the other is set to 8.

Infinite solutions

A system of equations has infinite solutions when both equations are essentially the same:

5) are all solutions to the system above, to name just a few. Note that

3x +2y=5
also has an infinite number of solutions. The first equation can be divided by 2 to get the same equation as the
second.

EXAMPLE 2:

ny -€32,

In thé IfthesyS€em has infimtely many solutions, whåt¯

is thefvalue of tn + n?

Both equations need to be the same for there to be an infinite number of solutions. We multiply the first

equation by 4 to get the right hand sides to match:

12x- 20y 32
mx — ny = 32

Now we can clearly see that m = 12 and n 20. Therefore, m+n

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CHAPTER 9 SYSTEMS OF EQUATIONS

Word problems
You will most definitely run into a question that asks you to translate a situation into a system of equations.
Here's a classic example:

EXÅMPLE orderl fronya restaurant. Each stuåent gets either a burgeror

$5 salad is $6, •If the group spent @total 'O€$16?Ahow
nan sntdents ofaeredburgers?

number of
Let x be the students who ordered burgers and y be the number who ordered salads. We can then
make two equations:

x +Y 30
5x +6y — 162

Make sure you completely understand how these equations were made. This type of question is guaranteed
to be on the test.
We'll use elimination to solve this system. Multiply the first equation by 6 and subtract:

6x -F 6y 180

5x + 6y = 162

18

18 students got burgers.

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 THE COLLEGE PANDA

More complex systems

You might encounter systems of equations that are a bit more complicated than the standard ones you've seen
above. For these systems, substitution and some equation manipulation will typically do the trick.

EXAMPLE 4:

(x, y) IS a solution io the systemo%f equations above and y > O; whatis the valueofy?

In the first equation, we isolate y to get y —3x. Plugging this into the second equation,

76

x2 + 2(9x2) — 76

x2 + 18x2 76

19x
2 — 76
x2

If x 2, then y -3(2) —6. If x —2, then y -3(-2) — 6. Because y >

EXAMPLE S:

2y
2

if'(x, y) ié a solution*o the equationabovepwhat wæpossiblevalffe for [VI?

+ 2)'s lying around in both equations. This is a hint that there might be a clever substitution
Notice the (x
somewhere, especially for a problem as complicated as this one. Isolating y in the first equation,

1 1
From here, Why would I want this form? So I can substitute in the second equation with
2
As you do these tougher questions, you must keep an eye out for any simplifying manipulations such as this
one.

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CHAPTER 9 SYSTEMS OF EQUATIONS

Substituting, we get
2
1 1
-6—0
2

2 2

-6=0
24—0
—o

Finally, y — —6 or 4, and yl can be either 6 or 4 .

How will you know whether there's a clever substitution or "trick" you can use? Practice. And even then,
you won't always know for sure. Just keep in mind that SAT questions are designed to be done without a
crazy number of steps. So if you feel like you're running in circles or hitting a wall, take a step back and try
something else. To get a perfect score, you must be comfortable with trial and error.

Here's the trick. Multiply all three equations. Multiply the left sides, and multiply the right sides. The result
is

x2y2z2 = 400
Square root both sides.

xyz =
Notice how we were able to get the answer without knowing the values of x, y, or z.

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Graphs
Learning a bit about equations and their graphs will inform our understanding of systems of equations.

The solutions to a system of equations are the intersection points. Therefore, the number of solutions to a
system of equations is equal to the number of intersection points.

Take, for example, the system of equations at the beginning of this chapter:

—3x — 2y -12

You can change both equations into y mx + b form (we won't show that here) to get the following lines.

The solution to the system, (2, 3), is the intersection point. There is only one intersection point, so there is only
one solution.

What about graphs of systems that have infinite solutions or no solutions?

Graphing the following system, which has no solution because its equations contradict each other,

—1

we get

What do you notice about the lines? They have no intersection points. They're parallel. Makes sense,
right?

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CHAPTER 9 SYSTEMS OF EQUATIONS

And for a system with infinite solutions?

2y
y

It's just one line! Well, actually it's two lines, but because they're the same line, they overlap and intersect in
an infinite number of places. Hence, an infinite number of solutions.

'EXAMPLE 7:

A system b oequatiOpSänd theirgraphs in the xy+ane are shown above. How many Solutions(does
thesystqn
C)Three D)Foue

Simple. The graphs intersect in two places so there are two solutions. Answer (B)

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CHAPTER EXERCISE: Answers for this chapter start on page 272.

A calculator should NOT be used on the

following questions.
2x +5y 24
x 4y 15

-11
If (x, y) satisfies the system of equations above,
x—i-3y what is the value of x + y?

What is the solution (x, y) to the system of

equations above?

A)
B)
D) 10

C)
D)

3x +Y
-10
y + 2x— 20 If (x, y) is a solution to the system of equations
6x -5y — 12 above, what is the value of xy?

A) —16
What is the solution (x, y) to the system of
equations above?

D) (7,6)

3x -4y 21
— — 14

If (x, y) is a solution to the system of equations

above, what is the value of y — x?

A) -18

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CHAPTER 9 SYSTEMS OF EQUATIONS

Y x2 +1 2x 4y=8

How many solutions (x, y) are there to the

system of equations above?

A) Zero
B) One
C) Two
D) More than two
x

2x — 5y = a
A system of two equations and their graphs in —8
the xy-plane are shown above. How many
solutions does the system have?
In the system of equations above, a and b are
A) Zero constants. If the system has infinitely many
solutions, what is the value of a?
B) One
C) Two
D) Three 1

D) 16

2(2x -1) = 3-3y 10

What is the solution (x, y) to the system of

equations above?
3x -6y — 20

In the system of equations above, a is a constant.

B) (-1,3)
If the system has one solution, which of the
following can NOT be the value of a?
D) (3, -17)
1

3
4

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 THE COLLEGE PANDA

11 14

1 3x - 61/ 15

—2x + 4y -10
— 4 x + 16
How many solutions (x, y) are there to the
What is the solution (x, y) to the system of system of equations above?
equations above?
A) Zero
A) (-2,8) B) One
B) (-1, 12) C) Two
C) (1, 20) D) More than two
D) (3, 28)

15
12
mx —6y — 10
y o.5x + 14 2x ny =5
-18
In the system of equations above, m and n are
According to the system of equations above, constants. If the system has infinitely many
what is the value of y?
solutions, what is the value of —?
n
1

12
1

3
4
3
13

1 1
—x
3 6 16

6x — ay 8

In the system of equations above, a is a constant.

If the system has no solution, what is the value
of a?
If (x, y) is the solution to the system of equations
1 above, what is the value of y?
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CHAPTER 9 SYSTEMS OF EQUATIONS

19

A calculator is allowed on the following

questions.

A local supermarket sells jelly in small, medium,

and large jars. Sixteen small jars weigh as much
as two medium jars and one large jar. Four small
A game of darts rewards points depending on
jars and one medium jar have the same weight
which region is hit. There are two regions, A and
asone large jar. How many small jars have the
B, as shown above. James throws 3 darts, hitting
weight of one large jar? region A once and region B twice, for a total of
18 points. Oleg also throws 3 darts, but hits
regions A twice and region B once for a total of
21 points. How many points are rewarded for
hitting region B once?
D) 10

18

On a math test with 30 questions, 5 points are

rewarded for each correct answer and 2 points
are deducted for each incorrect answer. If James
answered all the questions and scored 59 points,

solving which of the following systems of

equations gives his number of correct answers, A restaurant has two types of tables, rectangular
x, and his number of incorrect answers, y, on the ones that can each seat 4 people and circular
math test? tables that can each seat 8 people. If 144 people
are enough to fill all 30 tables at the restaurant,
¯ 59 how many rectangular tables does the restaurant
have?
30
A) 12

B)
— 30 B) 16
59 C) 20
D) 24
— 30
2x — 5y 59

30
5x — 2y 59

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 Inequalities
Just as we had equations and systems of equations, we can have inequalities and systems of inequalities.
The only difference is that you must reverse the sign every time you either multiply or divide both sides by a
negative number.

For example,

Do we have to reverse the sign at any point? Well, we would subtract by 3 to get 2x < 6 and then divide by
2 to get x < 3. Yes, we did a subtraction but at no point did we multiply or divide by a negative number.
Therefore, the sign stays the same.

Let's take another example:

The first step is to combine like terms. We subtract both sides by 4x to get the x's on the left hand side. We then
subtract both sides by 5 to get the constants on the right hand side:

3x — 4x < 4 —5

Notice that the sign hasn't changed yet. Now, to get rid of the negative in front of the x, we need to multiply
both sides by —1. Doing so means we need to reverse the sign.

This concept is the cause of so many silly mistakes that it's important to reiterate it. Just working with negative
numbers does NOT mean you need to change the sign. Some students see that they're dividing a negative
number and impulsively reverse the sign. Don't do that. Only reverse the sign when you multiply or divide
both sides by a negative number.

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CHAPTER 10 INEQUALITIES

EXAMPLE foÄiowmghntegersiSa 27?

27
4x < -20
—5

At no point did we multiply or divide by a negative number so there was no need to reverse the sign. We
divided a negative number, —20, but we did so by a positive number, 4.

The only answer choice that satisfies x —5 is —6, answer (A)

•cEXAMPLE 2: If —7 < ISIwhich Of the foiü)wipghrustbe true?

So how do we solve these "two-inequalities-in-one" problems? Well, we can split them up into two inequalities
that we can solve separately:

-21+3 < 15

Solving the first inequality,

Solving the second inequality,

-2x +3 15

—2x < 12

Putting the two results together, we get —6 x 5. Answer (C) .

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 THE COLLEGE PANDA

folJOyv$i$ diecplan, Jamé$ his daily sugai cOnstimptiOh to 408ramS,

Che cookie has 5 grams of sugar and one fruit salad contains7 grams of sugar; IfJames ate onlysookies
and fruit salads, which of the following: Inequalities.represents the possible number ofcookmes c and fruit
salåds s IthatøGcouideat Inone day and

A) eno 0.4.7 < 40

45
< 40

The total amount of sugar he gets from cookies is 5c. The total amount of sugar he gets from fruit salads is 7s.
So his total sugar intake for any given day is 5c + 7s, and since it can't be more than 40 grams, 5c + 7s < 40.

Answer (D) .

From a graphing standpoint, what does an inequality look like? What does it mean for y >
Y

As shown by the shaded region above, the inequality y > —x — 1 represents all the points above the line
x — you have a hard time keeping track of what's above a line and what's below, just look at the
1. If

y-axis. The line cuts the y-axis into two parts. The top part of the y-axis is always in the "above" region. The
bottom part of the y-axis is always in the "below" region. If the graph doesn't show the intersection with
the y-axis, you can always just draw your own vertical line through the graph to determine the "above" and
"below" regions.
Also note that the line is dashed. Because y > —x — 1 and NOT y x — 1, the points on the line itself do
not satisfy the inequality. If the equation were y > x 1, then the line would be solid, and points on the line
would satisfy the inequality.

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CHAPTER 10 INEQUALITIES

But what about a system of inequalities?

When it comes to graphing, the goal is to find the region with the points that satisfy the system. In this case,
1
we want all the points below y — x + 4 but above y — —x — 3. We can shade the regions below y —
2
1
and above y —x — 3 to see where the shaded regions overlap. The overlapping region will contain all the
2
points that satisfy both inequalities.

The overlapping region on the left represents all the solutions to the system.

Setting the two equations equal to each other and solving gives the intersection point, which, in this case,
happens to be the solution with the highest value of x.

— 14

4.66
—3

At x 4.66, y — —4.66 4 0.66 (we get this from the first equation). Therefore, (4.66, —0.66) is the solution
with the highest value of x. There are no solutions in which x is 5, 6, or larger.

While finding the intersection point in this example may have seemed a bit pointless (haha!), these points can
be very in the context of a given situation, such as finding the right price to maximize profit or
figuring out the right materials for a construction project.

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EXAMPLE 4;

111

The following system.of Inequalities IS graphed m the xy-piane above.

Which quadrants contain solutions to the system?

A) Quadrants I and II B) Quadrants J and IV C) Quadrants 111 and IV D) Quadrants I, 1!, and

First, graph the equations, preferably with your graphing calculator. Then shade the regions and find the
overlapping region.

*iiiF

As you can see, the overlapping region, which contains all the solutions, is the top region. It has points in

quadrants I, II, and IV. Answer (D)

EXAMPLE 5: Ecologiscs have determined that the number of frogs y must be greater than or equal to three
times the number of snakes x for a healthy ecosystem to be maintained m a particular forest. In addition;
the number of frogs and the number of snakes must sum to at -least 400.

PART 1: Whith of the following systems of inequalities expresses these conditions for a healthy ecosystem?

400 y + x. > 400 .y+x < 400

PART 2: What is the minimum possible number of frogs in a healthy ecosyytem?

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CHAPTER 10 INEQUALITIES

Part 1 Solution: The number of frogs, y, must be at least three times the number of snakes, x. So, y 3x. The
number of frogs and the number of snakes must sum to at least 400, so y + x > 400. Answer (C)

Part 2 Solution: In these types of questions, the intersection point is typically what we're looking for, but we'll
graph the inequalities just to make sure. First, put the second inequality into y mx + b form.

Y —x + 400

Then graph the inequalities using your calculator.

Y

The graph confirms that y, the number of frogs, is at a minimum at the intersection point. After all, the
overlapping region (the top region) represents all possible solutions and the intersection point is at the bottom
of this region, representing the solution with the minimum number of frogs.

We can find the coordinates of that intersection point by solving a system of equations based on the two
lines.

Substituting the first equation into the second,

400
x 100

100 is the x-coordinate. The y-coordinate is y 3x 3(100) 300. The intersection point is at (100, 300) and
the minimum number of frogs is in a healthy ecosystem.

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 THE COLLEGE PANDA

CHAPTER EXERCISE: Answers for this chapter start on page 275.

A calculator is allowed on the following

questions.
Y

Which of the following is a solution to the

inequality —x — 4 > 4x — 14?
x

Which of the following systems of inequalities

could be the one graphed in the xy-plane above?

A)
If —x —4 > x — 10, which of the following

must be true?
A) x < 24 B)

B) x > 24
C) x < -24
C)
D) x > —24

D)

Jerry estimates that there are m marbles in a jar.

Harry, who knows the actual number of marbles
in the jar, notes that the actual number, n, is

within 10 marbles (inclusive) of Jerry's estimate.

Which of the following inequalities represents
the relationship between Jerry's estimate and the
actual number of marbles in the jar?

A) n+10<mfn-10
B) m -10 10

C) n S m IOn

D) —
10
< n < 10m

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CHAPTER 10 INEQUALITIES

A manufacturer produces chairs for a retail store Y

according to the formula, M 12P + 100, where

M is number of units produced and P is the
the
retail price of each chair. The number of units

sold by the retail store is given by

N — —3P + 970, where N is the number of units
sold and P is the retail price of each chair. What
are all the values of P for which the number of
units produced is greater than or equal to the
number of units sold?

A) P > 58 The graph in the xy-plane above could represent

B) P f 58 which of the following systems of inequalities?
C) P 255 A)
D) PS 55
B)

If n is an integer and 3(n — 2) > what

is the least possible value of n?

To get to work, Harry must travel 8 miles by bus

and 16 miles by train everyday. The bus travels
at an average speed of x miles per hour and the
train travels at an average speed of y miles per
hour. If Harry's daily commute never takes more
than 1 hour, which of the following inequalities
represents the possible average speeds of the bus
and train during the commute?
8 16

16 8

8 16

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 THE COLLEGE PANDA

10

An ice cream distributor contracts out to two

differentcompanies to manufacture cartons of
ice cream. Company A can produce 80 cartons
each hour and Company B can produce 140
cartons each hour. The distributor needs to fulfill Which of the following graphs in the xy-plane
an order of over 1,100 cartons in 10 hours of could represent the system of inequalities above?
contract time. It contracts out x hours to
Company A and the remaining hours to
Company B. Which of the following inequalities
gives all possible values of x in the context of
this problem?

80
> 1, 100
x 10 x

B) 140x+80(10 x) > 1, 100 B)

C) 80x + 140(10 x) > 1, 100

D) 80x + 140(x- 10) > 1, 100

C)

D)

fözz:

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CHAPTER 10 INEQUALITIES

11 13

y > 15x +a If k x 3k + 12, which of the following must

y < 51 + b be true?
1. x- 12 < 3k
In the system of inequalities above, a and b are II. k > 6
constants. a solution to the system,
If (1, 20) is
111.
which of the following could be the value of
A) I only

B) I and II only
C) 11 and 111 only
D) 1, 11, and 111
C) 10
D) 12
14

12 20 9
If what is one possible
3 2'

Tina works no more than 30 hours at a nail salon value of x

each week. She can do a manicure in 20 minutes

and a pedicure in 30 minutes. Each manicure
earns her $25 and each pedicure earns her $40,
and she must earn at least $900 to cover her
expenses. If during one week, she does enough
manicures m and pedicures p to cover her
expenses, which of the following systems of
inequalities describes her working hours and her

earnings?
A) 3m + 2 < 30 Joyce wants to create a rectangular garden that
has an area of at least 300 square meters and a
25m + 40p 900
perimeter of at least 70 meters. If the length of

B) 2m + 3p < 30 the garden x meters long and the width is y

is

meters long, which of the follow systems of

25m + 40p 900
inequalities represents Joyce's requirements?

A) 70
C) + — < 30
x +y 300
25m + 40p 900
B) xy 150
D) + 900

25m + 40p < 30 C) xy2300

x 70

D) xy 300
x +y 35

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 THE COLLEGE PANDA

16

If a < b, which of the following must be true?

I. a2 < b2
11. 2a < 2b
Ill.

A) II only

B) I and II only

C) 11 and 111 only

D) 1, 11, and 111

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 Word Problems
For many students, solving word problems is a frustrating experience. They require you to translate the
question before you can even do the math. The examples and the exercises in this chapter will show you how
to handle the full range of word problems that are tested. You will develop an instinct for translating words
into math, setting the right variables, and finally solving for the answer. Experience is the best guide.

E PLE l•åhesumof threeconsecutive vers•s 72, What is the t ofthesé three integer

The most important technique in solving word problems is to let a variable be one of the things you don't
know. In we don't know any of the three integers, so we let the smallest one be
this problem, x. It doesn't
matter which number we set as x, as long as we're consistent throughout the problem.

So if x is the smallest, then our consecutive integers are

Because they sum to 72, we can make an equation:

72
— 72
3x — 69
x —23

And because x is the smallest, our three consecutive integers must be

23, 24, 25

The largest one is

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 THE COLLEGE PANDA

What if we had let x be the largest integer? Our three integers would've been

And our solution would've looked like this:

72
3x -3 — 72

75
x — 25

And because x was set to be the largest of the three integers this time, we're already at the answer!
On SAT word problems, think about which unknown you want to set as a variable. Often times, that unknown
will be what the question is asking for. Other times, it will be an unknown you specifically choose to make
the problem easier to set up and solve. And sometimes, as was the case in Example 1, it doesn't matter which
unknown you pick; you'll end up with the same answer with the same effort.

EXAMPLE(2•, One number is times anothegtnumberv Ifth y Sunyt044, what the larger of the two
numbers?

In this problem, we want to set x to be the smaller of the two numbers. That way, the two numbers can be
expressed as
x and 3x

If we let x be the larger of the two, we would have to work with

x
x and
3

and fractions are yucky.

Setting up our equation,

x + 3x 44
44
x —11

Be careful—we're not done yet! The question asks for the larger of the two, so we have to multiply x by 3 to

get 33 .

EXAMPLE 3: 'What is a nümbevsuch that the. square of the numberigequal t02.7%B0fitsreciprocål?

Let the number we're looking for be x.

2 1
x .027 x
x

Multiply both sides by x to isolate it.

.027

Cube root both sides.

x .3

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CHAPTER 11 WORD PROBLEMS

In 5 ars Albert will betwice

is Albertoow?r..

Let x be Albert's age now. We could've assigned x to be Henry's age, but as we mentioned earlier, assigning the
variable to bewhat the question is asking for is typically the faster route. Now at this point, some of you might
be thinking of assigning another variable to Henry's age. While that would certainly work, it would only add
more steps to the solution. Try to stick to one variable unless the question clearly calls for more.
If Albert is x years old now, then Henry must be x — 7 years old.
Five years from now, Albert will be x + 5 and Henry will be x — 2 years old.

x +5 2(x — 2)

EXAM}LÉSjfke per yards 10

The problem already gives us a variable t to work with. We want to equate Jake's distance run with Amy's.
Jake's distance: 60t
Amy's distance: 120(10) + 20(t — 10)
60t 120(10) + 20(t - 10)
60t — 1,200 + 20t - 200
— 1, ooo
t = 25

After 25 minutes, they will have run the same distance.

EXAMPLE 6. pharmaceutical yeresearch equipmen€musibeoshäred among the scienftists

There.isonemxcroscope every 4t=sts one
thegg is a total pieges of rsearc€equrpment at this company,how many scientists
there?

Let x be the number of scientists. Then the number of microscopes is , the number of centrifuges is —, and
x
the number of freezers is

52

Multiply both sides by 12 to get rid of the fractions,

= 52-12
13x — 624
x — 48

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 THE COLLEGE PANDA

EXAMPLE 7: MarkandKéVinown and — Lori ownSthé$estofth€

books. •If Kevin owns Wmore books than how many books does Lori own?

Let x be the total number of books. Mark then has —x books and Kevin has —x books. Kevin owns 9 more than
Mark, so
1 1
—x
3 4

Multiplying both sides by 12,

3x — 108
x 108

The total number of books is 108. Mark owns x 108 27 books and Kevin owns x 108 36 books. Lori

must then own 108 — 27 — 36 —@books.

EXAMPLE 8vA•group of friends wants to split the costOf rentinbä cabin equally. Ifeach friend pays $130
tOoniiCch. If ea@h friend pays $120, tbeywill have $50 too little it

"toarentthe eabifi?

We have two unknowns in this problem. We'll let the number of people in the group be n and the cost of
renting a cabin be c. From the information given, we can come up with two equations (make sure you see the
reasoning behind them):

13011 - 10

12011 + 50 c

In the first equation, 130n represents the total amount the group pays, but because that's 10 dollars toomuch,
we need to subtract 10 to arrive at the cost of rent, c. In the second equation, 120n represents the total amount
the group pays, but this time it's 50 dollars too little, so we need to add 50 to arrive at c. Substituting c from
the first equation into the second, we get

120n + 50 130n 10

-IOn 60

So there are 6 friends in the group. And

130n 10 130-6 10 770

The cost of renting the cabin is 770

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CHAPTER 11 WORD PROBLEMS

iXAMPLE9; aja57P0/9 and therest•are redA-row many green Jålybeans

must bexemoved so that60%Of JellybeanS are green?
VAC

The answer is NOT 20. You can't just take 10% of the green jellybeans away because as you do that, the total

number of jellybeans also goes down. We first find that there are x 200 — 140 green jellybeans. We need to
10
remove x of them so that 60% of what's left is green:

green jellybeans left

60%
total jellybeans left

140 x 6
200 x 10

Cross multiplying,

10(140 - x) — 6(200
1,400 - lox — 1,200 — 6x
200 4x
x — 50

green jellybeans need to be removed.

EXAMPLE I(kA1togeffver, a. 120 baseballcards. David gives Robert one thim of his
and ff1enÄ0noreéar4s ednöwhaefive times as many cards.as Dåvid, Mow many card' did
"tiraveori9nally±

Solution 1: This question is really tough and tricky. When David gives Robert some of his cards, David loses
at the same time Robert gains. We could set a variable for David and another variable for Robert, but that
solution is a little messier (see Solution 2).

Instead, let's work backwards. If x is the number of cards David ends up with, then Robert ends up with 5x
cards. Because there are 120 cards altogether,

x + 5x — 120
6x = 120
x — 20

So David has 20 cards and Robert has 100 cards at the end. Let's rollback another transaction. David had
given Robert 10 cards. So before that happened, David must have had 20 + 10 = 30 cards. Rollback another
transactionand we see that David had given a third of his cards away to get down to the 30 that we just
calculated. Well if he had given away a third, then the 30 he had left must have represented two-thirds of the

cards he had at the start. Let d be the number of cards David had at the start.

2
-d 30
3

90
d 45

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 THE COLLEGE PANDA

So David had 45 cards at the start, which means Robert must have had 120 — 45—@cards at the start.

Solution 2: number of cards David

Let x be the starts with and y be the number of cards Robert starts with.

Here are the equations I would set up:

120

1
y + _x + 10 -x - 10)
3

Multiplying the second equation by 3,

— 120

x 30 15x 150

Shifting things over,

x 120
9x - 3y — 180

At this point, we can use substitution or elimination. I'm going to use substitution. From the first equation,
x 120 — y. Plugging this into the second equation,

9(120 - y) — 180

1080 - 12y 180

-12y

y 75

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CHAPTER 11 WORD PROBLEMS

CHAPTER EXERCISE: Answers for this chapter start on page 277.

A calculator should NOT be used on the

following questions.
A retail store has monthly fixed costs of $3,000
and monthly salary costs of $2,500 for each
employee. If the store hires x employees for an
Which of the following represents the square of entire year, which of the following equations
the sum of x and y, decreased by the product of represents the store's total cost c, in dollars, for
x and y? the year?
A) x2 + Y2 — xy — 3, ooo + 2,
B) x2y2 — xy B) c— 2,500x)
C) c = + 2,500x
D) (x+y)2- xy D) c = 3,000

On a 100 cm ruler, lines are drawn at 10, X, and Susie buys 2 pieces of salmon, each weighing x
98 cm. The distance between the lines at X and pounds, and 1 piece of trout, weighing y
98 cm is three times the distance between the pounds, where x and y are integers. The salmon
lines at X and 10 cm. What is the value of X? cost $3.50 per pound and the trout cost $5 per
pound. If the total cost of the fish was $77, which
of the following could be the value of y?

If 5 is added to the square root of x, the result is

9. What is the value of x + 2? A calculator is allowed on the following
questions.

If 75% of 68 is the same as 85% of n, what is the

value of n?

A grocery store sells tomatoes in boxes of 4 or 10.

If Melanie buys x boxes of 4 and y boxes of 10,
where x and y 1 1, for a total of 60

tomatoes, what is one possible value of x?

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11

The Pirates won exactly 4 of their first 15 games. Ian has 20 football cards, and Jason has 44
They then played N remaining games and won baseball cards. They agree to trade such that
all they won exactly half of all the
of them. If Jason gives Ian 2 baseball cards for every card
games they played, what is the value of N? Ian gives to Jason. After how many such trades
will Ian and Jason each have an equal number of
cards?

B) 10

C) 11
D) 12

12
Alice and Julie start with the same number of
pens. After Alice gives 16 of her pens to Julie, If 3 is subtracted from 3 times thenumber x, the
Julie then has two times as many pens as Alice result is 21. What is the result when 8 is added to
does. How many pens did Alice have at the half of x?
start?
A) 1

D) 12

13
10
At a store, the price of a tie is k dollars less than

At a Hong Kong learning center, of the three times the price of a shirt. If a shirt costs $40
and a tie costs $30, what is the value of k?
students take debate, — of the students take

writing, and of the students take science. The

rest take math. If 33 students take math, what is

the total number of students at the learning

center?

A) 60
B) 66
C) 72
D) 78

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CHAPTER 11 WORD PROBLEMS

14

A bakery gave out coupons to celebrate its grand

opening. Each coupon was worth either $1, $3,
many $1 coupons were given out
or $5. Twice as
as $3 coupons,and 3 times as many $3 coupons
were given out as $5 coupons. The total value of
all the coupons given out was $360. How many

$3 coupons were given out?

A) 40
B) 45
C) 48
D) 54

15

Alex, Bob, and Carl all collect seashells. Bob has

half asmany seashells as Carl. Alex has three
many seashells as Bob. If Alex and Bob
times as
together have 60 seashells, how many seashells
does Carl have?

A) 15
B) 20
C) 30
D) 40

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 12 Lines
Lines are just functions in the form of f (x) = mx + b, which is why they are often referred to as linear functions.
We'll cover functions as a whole in a future chapter; we're covering lines first because they present some
concepts that don't apply to other functions. The SAT tests these concepts so frequently that they deserve their
own chapter. Let's dive in!

Given any two points (Xl, Yl) and (x2, Y2) on a line,

rise Y2 Yl
Slope of line
run X2 ¯ Xl

The slope is a measure of the steepness of a line—the bigger the slope, the more steep the line is. The rise is the
distance between the y coordinates and the run is the distance between the x coordinates. A slope of 2 means
2
the line goes 2 units up for every 1 unit to the right, or 2 units down for every 1 unit to the left. A slope of
means the line goes 2 units down for every 3 units to the right, or 2 units up for every 3 units to the left.

Lines with positive slope always go up and to the right as in the graph above.

rise

run
x

Lines with negative slope go down and to the right:

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 LINES

EXAMPLE 1:

The line shown in the xy-plane atxwe passes through the origin and point (a,b), where a > b. Which of
following could be the of the line?

rise
First, notice that the slope is positive. The slope, , is also equal to

b rise

a run

Since > b. is always less than L For example, if 5 and b 3, the slope would be The only choice

that's both positive and less than I is answer (B)

EXAMPLÉ 2: Linem passes through points 7) and (3 If the slope ofline m is 3Ävhat is the
value of k?

Slope
3-k

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 THE COLLEGE PANDA

Setting this equal to 3,

—3
k 11
k — 11 — 9 — 3k
4k — 20

EXAMPLE 3: If alinehas a slope of and passes through the point (1, which of the followingpomts
also lies on the line?

F) 1) P) (4, 10)

A slope of — means 1 up for every 3 to the right, or I down for every 3 to the left. If we go 3 to the left, the point
we get to on the line is( 2, —3). If we go 3 to the right, the point we get to on the line is (4, —1), answer (C) .

In this case, we got to the answer pretty quickly, but if we hadn't, we would have continued moving right or
left until we found an answer choice that matched. On the SAT, it shouldn't ever take too long to arrive at the
answer for a question like this.

In addition to slope, you also need to know what x and y intercepts are. The x-intercept is where the graph
crosses the x-axis. Likewise, the y-intercept is where the graph crosses the y-axis.

Let's say we have the line

- 12

To find the x-intercept, set y equal to 0.

2x + 3(o) — 12

12

The x-intercept is 6.

To find the y-intercept, set x equal to 0.

2(O) +3y — 12

3y = 12

The y-intercept is 4.

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CHAPTER 12 LINES

!XAMPLE4; Jf the line ax $3yea 15, wherea is constant; has an x-intercepiihat iS 9nce the Vålue of
is they@lue of a t'

First, set x = 0 to find the y-intercept:

a(o) +3y — 15

3y 15

The y-intercept is 5, which means the x-intercept must be 5 >< 2 10. Plugging in x = 10,y = o,
a(10) + 3(O) — 15
IOa — 15

a 1.5

All lines can be expressed in slope-intercept form:

y — mx + b

where m is the slope and b is the y-intercept. So for the line y 2x 3, the slope is 2 and the y-intercept is

—3:

x
2 1 o

While can be expressed in slope-intercept form, sometimes it'll take some work to get there. If you're
all lines

given a slope and a y-intercept, then of course it's really easy to get the equation of the line. But what if

we're handed a slope and a point instead of a slope and a y-intercept? Then it'll be more convenient to use
point-slope form:
y Yl m(x Xi)

where (Xl, Yl) is the given point. For example, let's say we want to find the equation of a line that has a slope
of 3 and passes through the point (1, —2). The equation of the line is then

Once it's in point-slope form, we can then expand and shift things around to get to slope-intercept form if we

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 THE COLLEGE PANDA

need to.

— 3(x —1)
y+2
y 3x —5

ÉXÅMPLÉS;

WhicWof the be the equation of fine shown inthe above?

To get the equation of the line y — mx + b, we need to find the slope m and the y-intercept b. The line crosses
the y-axis at 3, so b = 3. The line goes downward from left to right, down 1 for every 2 to the right, so the slope
1
m is —— . Therefore, the equation of the line is y — — —x + 3. Answer (C)
2

EXAMPLE 67A line I passa through the points ($2, 3)'and (O, 13b What is the ywintercept OfJin$: l?

Y2 ¯ Yl 13 3
Slope

Using point-slope form, our line is

y 13 20-3)
Note that we could've used the other point (—2, 3). The result will turn out to be the same.

Y 13 -3)
y 2x —6+ 13

After putting the equation into slope-intercept form, we can easily see that the y-intercept is Ø.

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CHAPTER 12 LINES

EXAMPLE Tifthelin

To find the point where two lines intersect, put them in y — mx + b form and set their equations equal to
each other. You're essentially solving a system of equations using substitution. In this case, the two lines are
already in y mx + b form. Setting them equal to each other,

31 -5 -2x + 10
5x — 15

When x = 3, y = 3(3) — 5 4. So the two lines intersect at (3, 4) and k

Two lines are parallel if they have the same slope.

Two lines are perpendicular if the product of their slopes is 1. In other words, if one slope is the negative

reciprocal of the other (e.g. 2 and —).

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 THE COLLEGE PANDA

EXAMPLEA: Lineåit has a slope of and passeéthrough,the (å,3) Af linen

m and passes through the same 3); which Of the following coul#e'he eqUation Of linep2

3
3 2

Because it's perpendicular to line m, line n must have a slope of . Using point-slope form,

3
—x +6+3
2
3
2

We get the equation into slope-intercept form to see that the answer is

Finally, you'll need to know the equations of horizontal and vertical lines. The equation of the vertical line that
passes through x 3 is, well, x 3.

The equation of the horizontal line that passes through (0, 3) is y

y —3

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CHAPTER 12 LINES

CHAPTER EXERCISE: Answers for this chapter start on page 279.

A calculator should NOT be used on the

following questions.
In the xy-plane, points (—3, 5) and (6, 8) lie on
line l. Which of the following points is also on
line l?
What is the equation of the line parallel to the
y-axis and 3 units to the right of the y-axis? A) (0,6)

—3
3 C) (9, 10)

—3 D) (12, 11)

EUäää" x

aasz:s
x
o

Note: Figure not drawn to scale. The graph of line I is shown in the xy-plane
above. Which of the following is an equation of
In the figure above, the slope of the line through
a line that is parallel to line I?

the two plotted points is —. What is the value of 2

3
2
3
3
2
7 3x — 1
y
3

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 THE COLLEGE PANDA

Y ax b
1
—bx

The equationsof two lines in the xy-plane are x

shown above, where a and b are constants. If the
two lines intersect at (2, 8), what is the value of

Line I in the xy-coordinate system above can be

represented by the equation y mx + b. Which
of the following must be true?
A) nib > O

B) mb<O
A calculator is allowed on the following C) nib —O
questions.
D) mb—l

m The y — —2x 2 is perpendicular to line l. If

line
these two lines have the same y-intercept, which
of the following could be the equation of line l?

2
What is the slope of the line m in the figure 1
above?
2

2
1
The slope of line I is — and its y-intercept is 3.
4
1
What is the equation of the line perpendicular to
line I that goes through (1, 5)?
2

—2x + 7
1 11
2 2
1 9
2 2

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CHAPTER 12 LINES

10

A line with a slope of passes through the

points (1, 4) and (x, 10). What is the value of x?

D) 10

11

If f (x) is a linear function such that f (2) f (3),

f(4) 2 f (5), and f (6) — 10, which of the
following must be true?

A) f(3) < f(o) < f(4)

B) f(o) =o
c) f(o) > 10
D) f(o) —

12

a
Y b

The equations of two perpendicular lines in the

xy-plane are shown above, where a, b, c, d, and e
are constants. If 0 < — < 1, which of the

following must be true?

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 Interpreting Linear
13
Models
On the SAT, you will encounter linear model questions that are a direct extension of the previous chapter
about lines. You'll have to interpret the meaning of the numbers in these models within a real world context,
applying your understanding of slope and y-intercept to do so.

1:ÄhevalueV;; In dollars/ of a .hGe from 2006 to 2015 can be estimated by the equation.
V —240000 T is thenumbér Ofyears Since 2006.

PART 13 Whichofthe followingbest describesthemeanin ofthenumber 240,000inthe equation?

A)- The value of the home in 2

*0B) valueof the home*im2015

average-value ofthe ho"e from 2006 to 20155

D) increase inthevalue•of the home #0012006 to 2015

PART 2: Which the following >est d+cribes of the number 58000 in the equation?

A)EThe number o!homes sold each year

B) The yearly dgcrease the value of dui'iome

9b The difference beyveen the value of and in 2015?

Dj the%yeariy décreasein tKevalue of thehome per square foot

Part 1 Solution: Many of these questions will give you an equation in y — mx + b form. The y-intercept b will
typically designate an initial value, the value when x = 0. In this case, the y-intercept is 240,000 and it describes

the value of the home when T = 0, zero years after 2006, which, of course, is 2006. Answer (A)

Part 2 Solution: Again, we're dealing with an equation of the form y — mx + b. The slope m always designates
a rate, the increase or decrease in y for each increase in x. In this case, the slope is —5, 000, which means the

value of the home decreases by 5,000 for each year that goes by. Answer (B)

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CHAPTER 13 INTERPRETING LINEAR MODELS

It's important that you don't get tricked into choosing a rate that looks right but ultimately doesn't fit the
context setby the variables x and y (in this case, T and V). Answer (A) is wrong because we're not dealing
with the number of homes sold; we're dealing with the value of a home. Answer (D) is wrong because the
numbers in the equation aren't on a per-square-foot basis. Always be aware of the variables you're working
with.

EXAMPLEæThema*iQunihéight in inches, c@be determined by the eqttåtionti

wherex Istheamountof•: •

1: Accordingtotheequation, on Å@regram0fferfilizer would increase the maximumheightofa

hop many inches?

EPART@To fieight&of exacåV0ne inch}, how many fertilizer

be gröwingtfie plant?r

Part 1 Solution: This question is essentially asking for the change in h for every 1 unit increase in x. This is

the slope. From the equation, we can see that the slope is —, or [O$] To make this even clearer, we can put the
4
equation into y mx + b fortn by splitting up the fraction: h — —x + —. Note that when we're dealing with
5
changes in x and y, the y-intercept b is irrelevant because it's a constant that's always there.

Part 2 Solution: Because this question is asking for the change in x for every 1 unit increase in h, the reverse of
Part 1, we need to rearrange the equation so that we have x in terms of h.

41 +6
5

5 3
—h
4 2

Now we can see that x increases by —, or@$l, when h increases by 1. The answer is just the slope of our new

equation. A shortcut for this type of question is to take the reciprocal of the slope of the original equation. The
reciprocal of — is

EXAMPLE

_Ä can OC90dæis put mto temperature mof the soda, indegrees VahrerOig$i ganbe 'found by
using wheregn is thenumber ofminutes the can Kay been üqareere is the m
decrease Ifie tentperature for everyS minutes the can is effin•the•'

The slope of —6 represents the change in the temperature for every 1 minute the can is left in the freezer. So

for every 5 minutes, the temperature of the soda decreases by 5 x 6 =@degrees Fahrenheit.

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 THE COLLEGE PANDA

CHAPTER EXERCISE: Answers for this chapter start on page 281.

A calculator should NOT be used on the

following questions.
A membership website offers video tutorials on
programming. The number of members, m,
subscribed to the site can be estimated by the
The water level h, in feet, in a large aquarium equation m = 500 + 200n, where n is the number
can be modeled by h 100 — 3d, where d is the of videos available on the site. Based on the
number of days that have passed since the equation, which of the following statements is
aquarium was last refilled. Based on the model, true?
how does the water level change each day?
A) For every one additional video, the site
A) Decreases by 3 feet gains 500 new members.
B) Increases by 3 feet
The site initially made 200 videos available
B)
C) Decreases by 100 feet to members.

D) Increases by 100 feet C) The site was able to get 500 members
without any available videos.
D) The site gains 500 new members for every
200 additional videos available on the site.

The number of loaves of bread b remaining in a

bakery each day can be estimated by the
equation b 200 18h, where h is the number
of hours thathave passed since the store's 10 2h
opening. What is the meaning of the value 18 in
A recipe suggests sweetening honey tea with
this equation?
sugar. The equation above can be used to
A) The bakery sells all its loaves of bread in 18 determine the amount of sugar s, in teaspoons,
hours. that should be added to a tea beverage with h

B) The bakery sells 18 loaves of bread each teaspoons of honey. What is the meaning of the 2
hour. in the equation?

C) The bakery sells a total of 18 loaves of bread A) For every teaspoon of honey in the
each day. beverage, two more teaspoons of sugar
D) There are 18 loaves of bread left in the should be added.

bakery at the end of each day. B) For every teaspoon of honey in the
beverage, two fewer teaspoons of sugar
should be added.

C) For every two teaspoons of honey in the

beverage, one more teaspoon of sugar
should be added.

D) For every two teaspoons of honey in the

beverage, one fewer teaspoon of sugar
should be added.

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CHAPTER 13 INTERPRETING LINEAR MODELS

The monthly salary of a salesperson at a used car h — 100 - 4t

dealership is determined by the expression The equation above can be used to model the
1, 000 + 2, 000xc, where x is the salesperson's
number of hours h until a gallon of milk held at
commission rate and c is the number of cars sold
a temperature of t, in degrees Celsius, goes sour.
by the salesperson. Which of the following Based on the model, which of the following is
statements is the best interpretation of the
the best interpretation of the number 4 in the
number 2,000 in the context of this problem?
equation?
A) The average price of a used car at the An increase of IOC will make a gallon of
A)
dealership milk go sour 4 hours faster.
B) The base monthly salary of a salesperson at
An increase of IOC will make 4 gallons of
B)
the dealership milk go sour 1 hour faster.
C) The average monthly commission earned An increase of 40 C will make a gallon of
C)
by each salesperson at the dealership milk go sour 1 hour faster.
D) The average number of cars sold by the An increase of 40 C will make a gallon of
D)
dealership each month milk go sour 4 hours faster.

p — 2,OOOS 15,000
An antique lamp was sold at an auction. The
A state government uses the equation above to price p of the lamp, in dollars, during the
estimate the average population p for atown auction can be modeled by the equation
with s Which of the following best
schools. p = 900 10t, where the number of seconds
t is

describes the meaning of the number 2,000 in the left in the auction. According to the model, what
equation? is the meaning of the 900 in the equation?
A) The average number of students at each A) The starting auction price of the lamp
school in a town The final auction price of the lamp
B)
B) The average number of schools in each C) The increase in the price of the lamp per
town second
C) The estimated increase in a town's
D) The time it took to auction off the lamp, in
population for each additional school seconds
D) The estimated population of a town
without any schools

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 THE COLLEGE PANDA

Questions 11-13 refer to the following

y 1.30x 1.50
information.
A bank teller uses the equation above to
exchange U.S. dollars into euros, where y is the
euro amount and x is the U.S. dollar amount. Daily Profit (y)

Which of the following is the best interpretation 600

500
of the 1.50 in the equation?
400
300
A) The bank charges 1.50 euros to do the 200
currency exchange. 100
Cakes (x)
B) The bank charges 1.50 U.S. dollars to do the -100
-200
currency exchange.
-300
C) One U.S. dollar is worth 1.50 euros. -400
-500
D) One euro is worth 1.50 U.S. dollars. -600
-700

The relationship between the daily profit y, in

A calculator is allowed on the following dollars, of a bakery and the number of cakes sold
questions. by the bakery is graphed in the xy-plane above.

10

2x +9 11
5

The equation above models the time t, in What does the slope of the line represent?
seconds, it takes to load a web page with x A) The price of each cake
images. Based on the model, by how many B) The profit generated from each cake sold
seconds does each image increase the load time
C) The daily profit generated from all the
of a web page?
cakes that were sold

D) The number of cakes that need to be sold to

make a daily profit of 100 dollars

12

Which of the following is the best interpretation

of the y-intercept in the context of this problem?

A) The price of each cake

B) The cost of making each cake
C) The daily costs of running a bakery
D) The daily cost of making the cakes that
weren't able to be sold

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CHAPTER 13 INTERPRETING LINEAR MODELS

13 15

What does it mean that (5, 0) is a solution to the 21/ -x — 14

equation of the line?
Alice owns a pet frog but would like to add
A) The bakery needs to sell 5 cakes per day to turtles to the same tank. The local veterinarian
cover its daily expenses. uses the equation above to determine the total

B) Each cake must be sold for at least 5 dollars amount of water y, in gallons, that should be
to cover the cost of making it. held in the tank for x turtles to thrive alongside
Alice's frog. Based on the equation, which of the
C) It costs 5 dollars to make each cake.
following must be true?
D) Each day, the bakery gives the first 5 cakes
away for free. I. One additional gallon of water can support
two more turtles.
II. One additional turtle requires two more
gallons of water.
Ill. One more turtle requires an additional half
14
a gallon of water.

T= 56 + 5k A) II only

B) 111 only
To warm up his room, Patrick turns on the
heater. The temperature T of his room, in C) I and II only
degrees Fahrenheit, can be modeled by the D) I and Ill only
equation above, where h is the number of hours
since the heater started running. Based on the
model, what is the temperature increase, in 16
degrees Fahrenheit, for every 30 minutes the
heater is turned on? c— 1.5 + 2.5x
A local post office uses the equation above to
determine the cost C, in dollars, of mailing a
shipment weighing x pounds. An increase of 10
dollars in the mailing cost is equivalent to an
increase of how many pounds in the weight of
the shipment?

B) 2.5

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 Functions
A function is a machine that takes an input, transforms it, and spits out an output. In math, functions are
denoted by f (x), with x being the input. So for the function

f(x) - x2 +1
every input is squared and then added to one to get the output. It's important to understand that x is a
completely arbitrary label—it's just a placeholder for the input. In fact, I can put in whatever I want as the
input, including values with x in them:

f(2x) (2x)2 1

f (Panda) (Panda)2 +1

Notice the careful use of parentheses. In the first equation, for example, (2x)2 is not the same as 2x2. Wrap the
input in parentheses and you'll never go wrong.

EkÅM?LEA: If f(x) then what Gf no) f(2) Ef(3n

Just plug in the inputs.

f(o) +f(l) +f(2) + f(3)

10 + 21 + 32 43

1 +2+9+64
76

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CHAPTER 14 FUNCTIONS

LEE
lox +25
*ZFor Whåt value Of$i$ the funcüOn fabove und 7

Because we can't divide by 0, a function is undefined when the denominator is zero. Setting the denominator
to zero,

x
2
- lox +25 — O

EXAMPLEB: Iff(xA)— 6x focgC2j)?

Whenever you see composite functions (functions of other functions), start from the inside and work your way
out. First,
g(2) = 2+3=5
Now we have to figure out the value of f (5).
Well, we can plug in x = 6 into f (x — 1) 6x to get f (5) 6(6)

g are jCg(j •hat IS

Again, we start from the inside and work our way out:

g(k+l) 2

2 2

Finally,

+1 10
2

k +1 18

17

As we've mentioned, a function takes an input and returns an output. Well, these input and output pairs allow
us to graph any function as a set of points in the xy-plane, with the input as x and the output as y. In fact,
y x2 + 1 is the same as f (x) — x2 + 1. Both f (x) and y are the same thing—they're used to denote the output.
The only reason we use y is that it's consistent with the y-axis being the y-axis.

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 THE COLLEGE PANDA

Anytime f (x) used in a graphing question, think of it as the y. So if a question states that f (x) > 0, all y
is

values are positive and the graph is always above the x-axis. It's extremely important that you learn to think

of points on a graph as the inputs and outputs of a function.

EXAMPLE 5;

The grkph of fÆ(xj is shown in the xy-plane above: For what valueof xis

Again, when it comes to graphs, interpret f (x) as the y. This question is asking for the point on the graph with

the highest y-value. That point is (5, 4). The x-value there is 5 .

XAMPLE 6; If the fundtio@with equation åx2 + 300sses the point (J, 2), what is the valUe Of g ,

Remember—a point is just an input and an output, an x and a y. Because (1, 2) is a point on the graph of the
function, we can plug in 1 for x and 2 for y.

EXAüPLh 7: If the function x2.+ ix •84 contams the pomt (m, im) and m What,ßihevalueofs»

It's important not to get intimidated by all the variables. The question gives us a point on the graph, so let's

plug it in.

2m = m +2m —4
0 tn2 —4
From here we can see that m— ±2. The question states that m > 0, so m 2

The zeros,xoots, and I-intercepts of mnctiowarealljustdifferentterms fOf the same

rof C"phicany, they refer tothe valåesoewherekhffuncåon rossesthe x-axis.

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CHAPTER 14 FUNCTIONS

EXAMPLE

aster

The graph X'SQ2x2 -g 6 IS shown intheå laneabOvec

P RT&Ä-iow manykdi'ånqtt

1...95

Part 1 Solution: The graph crosses the x-axis three times, so f has@distinct zeros. From the graph, we can
see that these zeros are —2, 1, and 3.

Part 2 Solution: This question is quite involved, so don't panic if you feel lost during the explanation. Read
all the way through and then go back to the bits that were confusing. I promise you'll be able to make sense of
everything.

To truly understand this question, first realize that a constant is just a function. No matter the input, we always
get the same output. In this question, we can write it as y k or g(x) — k. So let's say k -3. What does
—3 look like? A horizontal line at —3!

4
2
x
-1 O

Now when a question asks for the solutions to f (x) = k, it's merely referring to the intersection points of f (x)
and the horizontal line y = k. In general, if a question sets two functions equal to each other, f (x) g(x), and
asks you about the solutions, it's referring to the intersection points. After all, it's only at the intersection points
that the value of y is the same for both functions. In this particular case, g(x) just happens to be a constant
function, g(x) = k.

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 THE COLLEGE PANDA

The number of solutions is equivalent to the number of intersection points. So if k 3 as shown above, there
must be 3 solutions to f (x) 3, as represented by the 3 intersection points. The solutions themselves are the
x-values of those points. We can estimate them to be —2.2, 1.6, and 2.6.

Getting back to the original problem, we have to choose a k such that there is only one solution. Now we're
thinking backwards. Instead of being given the constant, we have to choose it. Where might we place a
horizontal line so that there's only one intersection point? Certainly not at —3 because we just showed how
that would result in 3 solutions.

Well, looking back at the graph, we could place one just above 8 or just below —4. Horizontal lines at these
values would intersect with f (x) just once. Looking at the answer choices, 9 is the only one that meets our

condition. Answer (D)

Let's take a moment to revisit part 1. In part 1, we found the number of intersection points between f (x) and
the x-axis. But realize that the x-axis is just the horizontal line y 0. In counting the number of intersection
points between f (x) and the horizontal line y 0, what you were really doing is finding the number of
solutions to f (x) 0.

If you didn't grasp everything in this example the first time through, it's 0k. Take your time and go through
it again, making sure you fully understand each of the concepts. The SAT will throw quite a few questions at
you related to the zeros of functions as well as the solutions to f (x) g(x).

Hopefully by now, you're starting to see constants as horizontal lines. So for instance, if f (x) > 5, that means

the entire graph of f is above the horizontal line y 5. Thinking of constants in this way will help you on a
lot of SAT graph questions.

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CHAPTER 14 FUNCTIONS

EXAMPJ.f9: Which of the followinwcouldbe thegraphofy.

O)

g:

Although the given function looks complicated and you might be tempted to graph it on your calculator, this
is the easiest question ever! All you have to do is find a point that's certain to be on the graph and eliminate

the graphs that don't have that point. So what's an easy point to find and test?

Plug in x 0 to get y 1. Now which graphs contain the coordinate (0, 1)? Only graph (B)

By the way, numbers like 0 and 1 are particularly good for finding "easy" points to use for this strategy.

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CHAPTER EXERCISE: Answers for this chapter start on page 283.

A calculator should NOT be used on the

following questions.
Y

x 2

o 20
1 21 34 3 2 4 5

3 29
—2
The table above displays several points on the
graph of the function f in the xy-plane. Which of 4
the following could be f (x)?

20x

B) f(x) x+20 The graph of the function f is shown in the

x 20 xy-plane above. If f (a) f (3), which of the
following could be the value of a?
X2 + 20
4

B) —3
2

D) 1

f(x)

g(x)
Y
x
O

In the portion of the xy-plane shown above, for

how many values of x does f (x) g(x)?

A) None
B) One
The function f is graphed in the xy-plane above.
C) Two
For how many values of x does f (x) = 3?
D) Three
A) Two
B) Three
C) Four
D) Five

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CHAPTER 14 FUNCTIONS

For which of the following functions is it true Iff(x) = x2, for which of the following values of

that f (—3) f (3)? c is f (c) < c?

2 1

x 2
3
x
3 3
2

10

The function f is defined by f (x) 3x + 2 and If the graph of the function f has x-intercepts at

the function g is defined by g(x) = f(2x) — 1. 3 and 2, and a y-intercept at 12, which of the
What is the value of g(10)? following could define f?

A) f(x)
B) f(x) — (x -4-3) (x —
C) f(x) (

D)

11

16 + x2 f(x) = x2 + 1
If f (x) — for all x 0, what is the value

of f (—4)?
The functions f and g are defined above. What is
the value of f(g(2))?

C) 10
D) 17

x 0 1 2
f(x) —2 3 18

Several values of the function f are given in the

table above. If f (x) ax2 + b where a and b are
constants, what is the value of f (3)?

A) 23
B) 39

C) 43
D) 56

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12

The function f is defined above for all x > 2.

Which of the following is equal to f (18) — f (11)?
A) f(3)
B) f(5)

c) f(6)
x D) f(7)
-3 o 3

A calculator is allowed on the following

The graph of the function f and line segment AB questions.
areshown in the xy-plane above. For how many
values of x between —3 and 3 does f (x) — 15

Which of the following points in the xy-plane is

NOT on the graph of y?

A)
13
B)

C)
x f(x)
3 D)
-2 5
2
16
2 16
3 4
Let the function g be defined by g(x) — 3x. If
4 8
6, what is the value of a?

The table above gives some values for the

function f. If g(x) 2f(x), what is the value of
k ifg(k) 8?

D) 12

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CHAPTER 14 FUNCTIONS

20

Questions 17-18 refer to the following

x f(x)
information.

x f (x) g(x) 1 —8
-2 3 4 2 3
-1 5 2 3 6
o —2 4 7
1 3 5 5 2
2 6 7 6 4
3 7 1
7 5

The functions f and g are defined for the six values Several values of the function f are given in the
of x shown in the table above. table above. If the function g is defined by
g(x) f(2x 1), what is the value of g(3)?
17

What is the value of f(g(—l))?

A) 2

21

f (x) 4x —3
18 g(x) = 3x+5

If g(c) 5, what is the value of f (c)?

The functions f and g are defined above. Which
of the following is equal to f (8)?
A) -2

B) g(3)

C) g(5)

19

If f(x) —3x 5 and —f (a) — 10, what is the

value of a?

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22 24

x
x 2 1 O i 2

—2
The graph of the function y — 9 — shown in
x2 is

the xy-plane above. What is the length of AB?

+ 1 is graphed in the
The function f (x) x3
xy-plane above. If the function g is defined by
B) 3 10
g(x) x + k, where k is a constant, and
f (x) — g(x) has 3 solutions, which of the
following could be the value of k?
D) 9 10
A) -1

23

25
2

x In the xy-plane, the function y ax + 12, where

0 1 a isa constant, passes through the point ( —a, a)
If a > 0, what is the value of a?

The function f is graphed in the xy-plane above.

If the function g is defined by g(x) f (x) + 4,
what is the x-intercept of g(x)?

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 Quadratics
Just as lines were one group of functions that have their own properties, quadratics are another. A quadratic
is a function in the form

in which the highest power of x is 2. The graph of a quadratic is a parabola.

To review quadratics, we'll walk through a few examples to demonstrate the various properties you need to
know.

QUAD TICIV

The Roots
The roots refer to the values of x that make f (x) = 0. They're also called x-intercepts and solutions. We'll
mainly use the term "root" in this chapter, but the other terms are just as common. Don't forget that they all

mean the same thing. Here, we can just factor to find the roots:

x2 -4x 21

The roots are 7 and —3. Graphically, this means the quadratic crosses the x-axis at x 7 and x = -3.

The Sum and Product of the Roots

We already found the roots, so their sum is just 7 + (—3) 4 and their product is just 7 x —3 21. This
was really easy, so why do we care about these values? Because sometimes you'll have to find the sum or the
product of the roots without knowing the roots themselves. How do we do that?

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b
Given a quadratic of the form y — ax2 + bx + c, the sum of the roots is equal to and the product of the roots
a

is equal to . In our example, a -21. so,

b —4
Sum
a 1

-21
Product — —21
a 1

See how we were able to determine these values without knowing the roots themselves? The roots that we
found earlier just confirm our values.

The Vertex
The vertex is the midpoint of a parabola.

x
-3 O 7

vertex

The x-coordinate of the vertex is always the midpoint of the two roots, which can be found by averaging them.

Because the roots are 7 and —3, the vertex is at x — 2. When x 2, f (x)
-21 — -25.
2
Therefore, the vertex is at (2, —25). Note that the maximum or minimum of a quadratic is always at the vertex.
In this case, it's a minimum of —25.

Vertex Form

Just as slope-intercept form (y mx + b) is one way of representing a line, vertex form is one way of representing
a quadratic function. We've already seen two different ways quadratics can be represented, namely standard
form (y = ax2 + bx + c) and factored form (y — (x — a) (x b)). Vertex form looks like y a(x + k.
To get a quadratic function into vertex form, we have to do something called completing the square. Let's walk
through it step-by-step:
y x2 — 4x — 21
See the middle term? The —4. That's the key. The first step is to divide it by 2 to get —2. Then write the

following:
2
21

See where we put the —2? The first part is done. Now the second step is to take that —2 and square it. We get
4.
2
- 21 —4
See where we put the 4? We subtracted it at the end. The vertex form is then

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CHAPTER 15 QUADRATICS

To recap, divide the middle coefficient by 2 to get the number inside the parentheses. Subtract the square of
that number at the end. Completing the square takes some time and practice, so if you didn't catch all of this,
first prove to yourself that it is indeed the same quadratic by expanding the result. Then repeat the process of

completing the square yourself. If you've been taught a slightly different way, feel free to use it. We'll do many
more examples in this chapter.
Now why do we care about vertex form? Well, look at the numbers! It's called vertex form for a reason. The
vertex —25) can be found just by looking at the numbers in the equation. But we already found the vertex,
(2,

you but we had to find the roots to do so earlier, and finding the roots is not always so
say! Yes, that's true,
easy. Vertex form allows us to find the vertex without knowing the roots of a quadratic. It's also very much
tested on the SAT!
2
One final note—one of the most common mistakes students make is to look at y (x 2) 25 and think the
vertex is at (—2, —25) instead of (2, 25). One pattern of thinking I use to avoid this mistake is to ask, What
value of x would make the thing inside the parentheses zero? Well, x 2 would make x 2 equal to 0. Therefore, the
vertex is at x — 2. This is the same type of thinking you would use to get the solutions from the factored form
Y— (x — a)(x b).

The Discriminant
If a quadratic is in the form ax2 + bx + c, then the discriminant is equal to b2 — 4ac. As we'll explain later, the
discriminant is a component of the quadratic forrnula. Before we explain its significance, let's calculate the
discriminant for our first example,
f (x) x2 — 4x — 21

Discriminant b — 100

Now, what does the discriminant mean? Well, the value of the discriminant does not matter. What matters is
the sign of the discriminant—whether it's positive, negative, or zero. In other words, we don't care that it's

100, we just care that it's positive. Letting D be short for discriminant,
When D > O, When D O, When D < O,

x
x
o
there are two real roots (two solutions). there is one real root. there are no real roots.

The Quadratic Formula

As we've seen, the roots are the most important aspect of a quadratic. Once you have the roots, things like
vertex form and the discriminant are not as helpful. Unfortunately, the roots aren't always easy to find or work
with. That's when vertex form, the discriminant, and the sum/ product of the roots can get us to the answer
faster.

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But if we must find the roots, there is always one surefire way to do so—the quadratic formula.

—b b2 — 4ac

for ax +c 0. For learning purposes, let's apply it to our example.

f(x) — x2 — 4x — 21

The roots or solutions are

4 ± 100 4 10
3
2(1) 2 2

And we get the same values as we did through factoring.

Notice that the discriminant, b2 — 4ac, is tucked under the square root in the quadratic formula. How does this
help us to understand what we know about the discriminant?
Well, when b2 — 4ac > 0, the and we end up with two different roots. When b2 — 4ac = 0, the
takes effect
"4" does not have an effect since we're essentially adding and subtracting 0, both of which give us the same
root. When b2 4ac < 0, we're taking the square root of a negative number, which is undefined and gives us
no real roots (we'll talk about imaginary number in a later chapter).

Hopefully, the quadratic formula helps you understand where the discriminant and its various meanings come
from. Understanding this connection will help you remember the concepts.
Now that we've taken you on a thorough tour through the properties of quadratics, we'll go through a few
more examples to illustrate some important variations, but we'll do so at a much faster pace.

:3.i0S:• •
QUADRAm1C

The Roots
This quadratic cannot be factored. And in fact, if we look at the discriminant,

b2 — 4ac

it's negative, which means there are no real roots or solutions. The graph of the quadratic makes this even
more clear:

x
O

When the coefficient of the x2 term is negative, the parabola is in the shape of an upside-down "U."

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CHAPTER 15 QUADRATICS

The Sum and Product of the Roots

f(x) - — X2 + 6X — 10

b 6
Sum
a —1
-10
Product — 10
a —1

Wait, what!? We already determined that there were no roots. How can there be a sum and a product of roots
have any real roots, but it does have imaginary roots. The values
that don't exist? Well, the quadratic doesn't
above are the sum and product of these imaginary roots. We'll cover imaginary numbers in a later chapter.

Vertex Form
Because the roots are imagmary, we can't use their midpoint to find the vertex. In these cases, we must get the
quadratic in vertex form. We'll have to complete the square.

— x2 + — 10

First, multiply everything by negative 1 to get the negative out of the x2 term. Having the negative there makes
things needlessly complicated. We'll multiply everything back by —1 later.

¯Y — x2 — 6x + 10

Divide the middle term by 2 to get —3 and square this result to get 9. Remember that we put the —3 inside the
parentheses with x and subtract the 9 at the end. Putting these pieces in place,

Now multiply everything by —1 again, 2

-0-3) -1
Now it's easy to see that the vertex is at (3, —1). And because the graph is an upside-down "U," —1 is the
maximum value of f (x).

QUAPRATIC

The Roots
We can factor this quadratic to get

2x2 + 5x -3 —o
—o
x = 05 3

The roots are 0.5 and —3. If you don't know how we factored this, unfortunately teaching factoring from the
ground up is not within the scope of this book. Don't be afraid to look up factoring lessons and drills online
and in your textbooks. It's an essential skill to have. Just know that all methods involve a little trial and error.

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If you're ever stuck, the quadratic formula is always an option.

The Sum and Product of the Roots

b 5
Sum -2.5
a 2

Product — -3
—1.5
a 2

The Vertex
Averaging the two roots to find the x-coordinate of the vertex,

0.5 + (—3) -2.5

-1.25
2 2

Plugging this into f (x) to find the y-coordinate,

f(-1.25) 1.25) —3 -6.125

The vertex is at (—1.25, —6.125). Because the quadratic opens upward in the shape of a "U," the maximum
value of f (x) is —6.125.

Vertex Form

y — 2x2 + 5x -3
First, divide everything by 2. Before completing the square, always make sure the coefficient of x2 is 1. We'll
multiply the 2 back later.

5 3
2 2

25
Divide the middle term by 2 to get and square this result to get —
16
We put the inside the parentheses with

25
x and subtract the —
16
at the end.

3 25
2 2 16

Combining the constants,

49
2 4) 16

Multiplying by 2,

49
45)2 8

y 2(x+ 1.25)2
- 6.125
This is consistent with the vertex found above.

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CHAPTER 15 QUADRATICS

The Discriminant
For the sake of completeness, let's calculate the discriminant. Hopefully, it will confirm the fact that this
quadratic has two distinct real roots.
y — 2x2 + 5x — 3

Discriminant — b2 — 4ac 49

The discriminant is positive, which confirms the fact that this quadratic has two real roots.

AUADRATIC4.—
— 121 +9

The Roots
We could factor this, but let's use the quadratic formula instead.

—b — 4ac 12) ( 12)2 — (9) 12 3

x
8 2

3
As you can see, the discriminant is 0 and the quadratic has just one root,

The Sum and Product of the Roots

b —12
Sum
a 4
c 9
Product

Ifwe only have one root, how is it that we can have a sum and a product of two roots? Why are they different
from the one root we found?

Here's the thing. While we may say a quadratic has just one root, it really has two roots that are the same.
After all, a quadratic, with an x term, is expected to have two roots. When they're the same, we just refer to
them as one.
9
So our "two" roots are — and —. If we add them, we do indeed get 3, and if we multiply them, we do get
4

The Vertex
When a quadratic has just one root, the x-coordinate of the vertex is the same as the root. That's because a
quadratic is tangent to the x-axis when it has one root.

x
o 3
2

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3
The y-coordinate is, of course, 0. Therefore, the vertex is at —,0 The minimum value of f (x) is 0.
2

Vertex Form

y 4x2 -12x+9
First, divide everything by 4. Before completing the square, always make sure the coefficient of x2 is 1. We'll
multiply the 4 back later.
2 9
— 3x +
4 4

3
Divide the middle term by 2 to get and square this result to get We put the inside the parentheses
2

with x and subtract the — at the end.

9 9
4 4 4

The constants cancel out.

2
3
x
4 2

Multiplying by 4,

32)2

This is consistent with the vertex found above.

Wow! We just covered pretty much everything you need to know about quadratics. Unfortunately, we're not
quite done yet as there are a few tough question variations that you should be exposed to.

EXAMPLE 1: In the xy-plane, the parabolæwithoequation y x intersects the lineu

attPoinV(m, b). Whatis thevalue of b?

Whenever you're finding the intersection of two graphs, you're essentially solving a system of two equations.
You're looking for an x and a y that will satisfy both equations. Typically, you just set the equations equal to
each other, much like we did when we were trying to find the intersection of two lines in the chapter on lines.
It's the same as substituting the second equation (the line) into the first equation (the parabola):

3x - 10 x — 5x 6

0 16

To find the y, we plug x 4 into either of the original equations: y 3(4) — 10 2. Therefore, the point of

intersection is at (4, 2) and b — a.

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CHAPTER 15 QUADRATICS

EXAMWE2: many does the of 6x'+3 lineu iO@y the

*y-plane?

We're dealing with intersections again! So what do we do first? Treat it as a system of equations. We plug the
second equation into the first.

10

Now we could go ahead and solve this to find the intersection point(s) like we did in the previous example, but
there's a faster way. For the purposes of this question, we don't care where the intersections are. We just want
to know how many there are.
Sound familiar? We can use the discriminant to do that.

Discriminant b2 — 4ac (6)2 - 8

The discriminant is positive, which means there are 2 solutions to the equation we set up above. If there are 2

solutions to the equation above, there must be intersection points. To summarize, we didn't bother finding
the two values of x. They could've been x 2 and x — 100 for all we care, and the intersection points might've
been (2, 5) and (100, 6). It doesn't matter. What mattered was that there were two of them, and we used the
discriminant to determine that. If the discriminant were 0, there would only be one intersection point. And if
the discriminant were less than 0, there would be no intersection points.

Make sure you understand this question. Feel free to go back and figure out where the intersection points
actually are (Hint: It's not fun. You'll need the quadratic formula. Be glad you know how to use the discriminant).

EXAMPLES

11 +
of equations a asonstant.yorcwhieh of $he following values Of & does the system
EEOf e4U@tions have no

First, we get y k from the first equation and substitute this into the second equation,

If the system of equations has no real solution, then the equation above should have no real solution. The
discriminant should be less than 0.

Discriminant — b2 — 4ac (-3)2 — 4(1)(1 — k) +4k _ 5 4k

Now we test each of the answer choices to see which one results in 5 + 4k being negative. Only —2, answer

(A) produces a negative discriminant.

These questions are some of the toughest you'll see on the SAT, especially when you can't use your calculator.
Go back and make sure you understand them.

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—EXAMPLE 4: A biOJogistUsestnefimction p(å) —100/324 modelthé population.of'eagulls

on a beach in year number n,wherel 'Which of thQToIlowing eqt11Valentf6rmsOf p(n) displays
10.
maximum populatåon of seagulls and thenumber of tKe Yfar in wi-åcn the population reacK6s that*'
'maximum as constants or coefficients?

4n(25n 250) 1000n?U100h) +2,50

(Dji(h)

Anytime you see a quadratics question with the words maximum or minimum, either figure out the vertex
or look for vertex form. After all, the vertex is where the maximum or minimum occurs. In fact, the answer is
either (C) or (D) because those are the only ones in vertex form. Furthermore, with a little calculation, it's easy
to see that (D) does not expand to be the original equation, so the answer is (C).

However, for learning purposes (and for the tougher questions), I'll show you how to do this question in two
different ways. We can find the vertex using the average of the roots and then reverse engineer the vertex form.
Or we can transform the equation into vertex form directly.
Solution 1: To find the roots, we set the equation equal to 0 and factor,

-IOOn2 + 1, OOOn O
-IOOn(n - 10) = O
n = O, 10

The roots are 0 and 10, which means the x-coordinate of the vertex is 5. Now we can plug 5 into p(n) to find
the y-coordinate.
p(5) + 1, 000(5) 2, 500

So the vertex is at (5, 2500). Now remember what vertex form looks like: y = a(x — + k. Given our values,
we have
p(n) - + 2, 500
We now need to find what a is. To do that, we need another point to work with. Well, it's easy to see that p(n)
passes through the point (0, 0). Plugging that in,

O - + 2,500
o 25a + 2, 500
-25a — 2,500

a -100

Finally, p(n) — -100(n - + 2,500. Answer (C) .

Solution 2: This second method involves completing the square to get the vertex form directly. First, divide
everything by —100 to ensure the coefficient of 112 is 1.

p(n) — -IOOn2 + 1, ooon

p(n) 2
n 10n
-100

Do you remember what to do next? If we wrote the constant 0 at the end, the "middle" term would be — Ion.
Divide the —10 by 2 to get —5 and square that to get 25. The —5 belongs inside the parentheses with n and the

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CHAPTER 15 QUADRATICS

25 gets subtracted at the end.

p(n) 2
25

Now we can multiply everything back by —100.

p(n) — -100(n - + 2,500

And again, we prove that the answer is (C)

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Reviewee
Given a quadratic Of the form, y ax2 + bx +

The also called $01utions and x-intercepts, can be .found in the following ways:
• Factoring

• Graph on the calculator (look for the X-intercepts)

—0 .02 4ac
• The quadratic formula x
2a,

b
Sum of the Roots
a

Product of the Roots

a

The discriminant D — b? 4ac

• 'When,D > 0; there are tworeal solutions.

• When D 0, there isone real solution.

• When D there areno real solutions.

To find the vertex,

Take the average of the roots to get the x-coordmate. Then plug that value into the quadratic to get"
the y-coordinate€

e Put the quadratic in vertex form by completing the square.

l. $,nsure thecoefficment of 12 IS positive I bydividing everything by a.

a a

b2
2. Divide thecoefficient Of the middle term to get Square that result to get Put
4a2 Oa
inside the parentheses wi$h x and subtract 4a2 at thé end

2
b c b2
a 2ti a 4a2

3. Multiply everything by a.

2a

4. It's unnecessary to memorize these steps with the variables. Practice on quadratics withActual
numbers. uowever, do remember what vertex form looks like: y a(x — +

Whenever you're asked for the minimum orthe maximum of a quadratic, find thewertex,

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CHAPTER 15 QUADRATICS

CHAPTER EXERCISE: Answers for this chapter start on page 286.

A calculator should NOT be used on the

following questions.
3x2 + lox = 8

Ifa and b are the two solutions to the equation

above and a > b, what is the value of b2?
In the xy-plane, what is the distance between the
two x-intercepts of the parabola 4
9
2
3

D) 16
D) 10

What is the sum of the solutions of

What are the solutions to x2 + 4x + 2 0?

B) x=2+2Vi

6
If a < 1 and 2a2 7a +3 0, what is the value
Of a?
—3

In the system of equations above, c is a constant.

For which of the following values of c does the
system of equations have exactly two real
solutions?

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 THE COLLEGE PANDA

10

P - 100m 120, OOO

A calculator is allowed on the following
questions. The monthly profit of a mattress company can
be modeled by the equation above, where P is
the profit, in dollars, and m is the number of
mattresses sold. What is the minimum number
At which of the following points does the line of mattresses the company must sell in a given
with equation y 4 intersect the parabola month so that it does not lose money during that
y (x + — 5 in the xy-plane? month?
A) (—1,4) and (—5, 4)
B) (1,4) and (-5,4)
C) (1,4) and (5,4)
D) ( 11,4) and (7,4)

11

—3

y ax +4x—4
In the system of equations above, ais a constant.

For which of the following values of a does the

x system of equations have exactly one real
0 1 5 6
solution?

A)
B) -2
Which of the following equations represents the
parabola shown in the xy-plane above?

A) y — (x 3)2-8
B) y — (x +3)2 +8
12
8

f(x) - —x2 + 6x + 20
The function f is defined above. Which of the
following equivalent forms of f (x) displays the
maximum value of f as a constant or coefficient?

For what value of t does the equation v 5t — A)

result in the maximum value of v? B) —(x — + 29
C) f(x) — -(x + 11
D) f(x) — (x + 29

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CHAPTER 15 QUADRATICS

y = a(x — 3)(x — k)
In the quadratic equation above, a and k are
constants. If the graph of the equation in the
xy-plane is a parabola with vertex (5, —32), what
is the value of a?

14

In the xy-plane, the line y = 2x + b intersects the

parabola y = x2 + bx + 5 at the point (3, k). If b
is a constant, what is the value of k?

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 Synthetic Division
Synthetic division involves dividing one polynomial by another in the same way you divided numbers in 3rd

grade.

18 R 2 x2 + 3x 2

356 x 1 x3 + 2x2 5x + 1

I'll teach you the long "mathematical" way first, but then direct you towards several shortcuts that will get

you through almost any synthetic division question on the SAT without using the long way. These questions
rarely show up, and if they do, they'll show up only once.

Let's retrace the steps of dividing 56 by 3 so you can see how the same logic applies to synthetic division.
First, we see that 3 goes into 5 once. We put a I on top and a 1 x 3 3 below the 5. We then subtract to get 2
and bring the 6 down.

356
3
26
Now how many times does 3 go into 26? 8 times. So we put an 8 up top and a 3 x 8 24 below the 26.
Subtracting, we get 2.
18
356
3
26
24
2

At this point, there are no more digits to bring down and 3 does not go into 2. Therefore, 3 goes into 56 eighteen
times with a remainder of two. This result can be written in the following form:

56 2
18
3

where 18 is the quotient, 2 is the remainder, and 3 is the divisor.

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CHAPTER 16 SYNTHETIC DIVISION

The process of dividing a polynomial is essentially the same. To show you how synthetic division works, let's
divide x3 + 2x2 — 5x + 1 by x — 1.

How many times does x — 1 go into x3? x2 times. Why? Because x x x2 = x3. The goal is to match x3. We

don't care about the —1 during this "fitting in" step. Now, (x — 1) x x2 x3 — x2. This is what we put below
the dividend.

x2

x —1 x3 + 2x2 5x + 1

x3 x 2

Finally, we subtract like we do in basic number division. Notice that we must subtract each element, so the —x2
becomes +x2, yielding 3x2. Unlike in long division with numbers, all the remaining terms from the dividend
should be brought down for each step in synthetic division.

x2

x I 2x2 5x + 1

x3 x2

3X2 5x + 1

Next step. How many times does x — 1 go into 3x2? 3x times. Remember our goal at each step is to get
the same exponent and the same coefficient as the term with the highest power. We put the +3x up top and
3x x (x 1) 3x2 — 3x on the bottom.

x2 + 3x
x I x3 2x2 5x + 1

3x2 5x + 1

3x2

And just like last time, we subtract each term, not just the first. We then bring down the 1.

x2 + 3x

x —1 x3 + 2x2 5x + 1

x3 x2

312 5x + 1

3 X2 3x

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 THE COLLEGE PANDA

We're almost done. How many times does x — 1 go into . —2 times. So a —2 goes up top and
—2x + 2 goes on the bottom.

x + 3x 2

X3 + 2x2 5x + 1

x3 x2

3X2
2

Subtracting, we get —1 at the end.

3x 2
2
x 5x 1

3 2
x x

3x 2 1

3X2 3x

56
We know we're done when we end up with a constant. And just as we can express as 18 , a mixed fraction,
3
we can express
x3 + 2x2 1
as x2 + 3x — 2

Notice where each component is placed. The quotient is written out in front. The remainder, —1, is the

numerator of the fraction and the divisor, x — 1, is the denominator. These placements are exactly the same as
in long division with actual numbers. Get used to seeing synthetic division results in this format.

Here's another thing that's the same. The result of our long division with numbers

18 R 2
356
means that 56 =3 x 18 -I- 2.

The same meaning applies to our synthetic division result.

x3 +2x2 -5x +1 —1
Dividend Quotient >< Divisor + Remainder

Hopefully you've been able to grasp synthetic division more intuitively through the comparison with regular
long division. All the parts relate to each other in the same way. Let's dive into some more examples where
we can show you some shortcuts.

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CHAPTER 16 SYNTHETIC DIVISION

EXAMPLE 1: Theexpresion equivalent to whichofthefollowing?

Using synthetic division,

6x + 12

17

17
The quotient is 6 and the remainder is — 17. We can write this result as 6 Answer (A)

Now how would we approach this question without using synthetic division?

We can plug in numbers that we make up. Let's say x — 2. Then -5

6(2) 7
2+2
We now look for an answer choice that gives — when x — 2. We can rule out (C) and (D) right away since they

don't give —. Plugging x 2 into answer (A) gives

17 17 24 17 7
6 6
4 4 4 4

This confirms that the answer is indeed (A). This strategy of making up numbers and testing each answer
choice can be much faster than synthetic division.

When +4 diviÅed by x result is What is A inffermspfxh

Using synthetic division,

3x2 + 4

3x + 4

If you followed along, you should've noticed it got a little clunky when we subtracted the —3x and brought
the 4 down. That's because the dividend, 3x2 -l- 4, has no x terrn. Still, the process is the same: subtract and
bring the remaining terms down.

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 THE COLLEGE PANDA

3x2 +4 7
The quotient is 3x + 3 and the remainder is 7. The result can be expressed as 3x +3+ . Now
it's easy to see that A 3x + 3, answer (C) .

Again, we could've done this question by making up numbers. If x 2, then

3x2 +4
— 16

we didn't know the answer was (C), we would test each answer choice with x
If 2 until we got 16, but since
we do know, we'll test (C) first for confirmation. Letting A 3x + 3,

7 7
3(2) 3 — 16

Answer confirmed.

jf thé txpress!on is written m the form ———„-nwhere B i? a •Cönstant,

•what Of

Based on where it is, B represents the remainder of the division.

5x + 6
2
1

5x2 lox

6x 12

13

13
We can write the result of this division as 5x + 6 + from which B

This last example is perfect for demonstrating a shortcut called the remainder theorem, which allows us to get
the remainder without going through synthetic division.

In Example we divided 5x2 — 4x + 1 by x — 2. Whenever a polynomial is divided by a monomial, which is

3,

just something in the form of ax + b, the remainder can be found by plugging in to the polynomial the value
of x that makes the monomial equal to 0. The process sounds more complicated than it is, so let's show how
it's done.

What makes x — 2 equal to 0? x 2.

Plug that into the polynomial 5x2 — 4x + 1.

13

And that's the remainder we obtained in Example 3.

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CHAPTER 16 SYNTHETIC DIVISION

What is the remainder when —2x2 -E 5x is divided by x -E 1?

Well, what makes x -F 1 equal to zero? x — 1. Plugging that into the polynomial,

Boom. —7 is the remainder.

What is the remainder when 4x4 + 3x2 — 4 is divided by 2x — 1?

1
What makes 2x — 1 equal to zero? x —
2

Plugging that into the polynomial,

4 2
1
4 -3
2

Boom. —3 is the remainder.

EXAMPLE If theé*pressipn is written in the equivalGt form 2x41 , what is thew•

ue

R represents the remainder after dividing 2x2 — 5x + 1 by x — 3. Using the remainder theorem, we can plug in
x — 3 into 2x2 — 5x + 1 to get the remainder.

- 5(3) +1 18 -15+1 4

No need for synthetic division.

One last thing about the remainder theorem. Let's say that we divide x2 — 3x + 2 by x 2. Plugging in 2, we
see that the remainder is

(2)2

Since the remainder is 0, x — 2 is a factor of x2 — 3x + 2, just like 3 is a factor of 18. And indeed, if we factor
x2 — 3x + 2,
(x — 2)(x 1)

we see that x — 2 is in fact a factor.

Don't you just love how everything in math is connected?
Some questions now become much easier. For example, is x -F 1 a factor of x3+ 1?
Well, plugging in —1, we find that the remainder is (—1)3 + 1 = 0. Therefore, x + 1 is a factor of x3 + 1.

Do note that the remainder theorem only works when we're dividing by monomials like x + 1. If we were
dividing x3 + 1 by something like x2 + 2, we would have to use synthetic division. Fortunately, the SAT will
never ask you to do that.

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 THE COLLEGE PANDA

•re(x) — —
In the polynomial fCx) defmed above, k •ns.a. consianå. Iff (i) is divisible by x—2, what isthe value of k?

If f (x) is divisibleby x 2, then the remainder is 0 when f (x) is divided by x — 2. In other words, x — 2 is a
factor of f (x). The remainder theorem tells us that when we plug 2 (the value that makes x — 2 equal to zero)
into f (x), we should get 0.

—o
—o
24 4k + 10 + 2 —o
36 4k —O
—36

Answer (B)

EXAMPLE 6.

The table aboye gives the value b/polynomialp(x) for some VAIue$ Of:x, Which

The remainder theorem makes this question easy. Because p ( 1) = 0, x -F 1 must be a factor of p(x). Answer
(A) . Be careful—the answer is NOT x — 1.

you found this chapter confusing, feel free to skip over it and come back. It's hard to make sense of synthetic
If

if you haven't encountered it before. It won't show up more than once, if at all, so don't let it keep
division
you from reviewing other topics.

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CHAPTER 16 SYNTHETIC DIVISION

CHAPTER EXERCISE: Answers for this chapter start on page 289.

A calculator should NOT be used on the

following questions.
x g(x)
-3 2

The expression is equal to which of the -2 3

0 —4
following?
1 —3
3 6
8
The function g is defined by a polynomial. The
8 table above shows some values of x and g(x).
C) What is the remainder when g(x) is divided by
D) 4-2x

If the expression is written in the

1
form + Q, what is Q in terms of x?

A) 3x-1
2z3 — kxz2 + 5xz + 2x — 2

C) 6x2 +3x+1 In the polynomial above, k is a constant. If z —1

is a factor of the polynomial above, what is the
D) 6x2 1
value of k?

The expression 4x + 5 can be written as

A (2x — 1) + R, where A is an expression in terms
of x and R is a constant. What is the value of R?

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 THE COLLEGE PANDA

10

A calculator is allowed on the following If p(x) x3 + x2 — 5x + 3, then p(x) is divisible

by which of the following?

questions.

II. x 1

111. X +3
When 3x2 — 8x — 4 is divided by 3x — 2, the
8 A) I and II only
result can be expressed as A What is A and only
3x—2• B) 1 111

in terms of x? C) II and Ill only

D) 1, 11, and 111

11

If the polynomial p(x) is divisible by x — 2,

which of the following could be p (x)?

A) p(x) — —x2 + 5x — 14
The expression 2x2 — 4x — 3 can be written as B) p(x) = x2 —6x—2
A(x + 1) + B, where B is a constant. What is A
in terms of x?
D) p(x) = 3x2 -2x -8

B) 2x+2 12

D) 2x-6 Ifx — 1 and x + 1 are both factors of the

polynomial ax4 + bx3 — 3x2 + 5x and a and b are
constants, what is the value of a?

The expression x2 + 4x — 9 can be written as

(ax + b)(x — 2) + c, where a, b, and c are
constants. What is the value of a + b + c?

13

D) 10 For a polynomial p(x), p 0. Which of the

following must be a factor of p (x) ?

A) 3x-1
B) 3x+1
For a polynomial p(x), p(2) 0. Which of the
following must be true about p (x)?

A) 2x is a factor of p(x).
B) 2x —2 is a factor of p(x).
C) x — 2 is a factor of p(x).
D) x +2 is a factor of p(x)

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 Complex Numbers
What value of x satisfies x2 —1? There were no values until mathematicians invented the imaginary
number i, which represents They defined i2 to equal —1, and from there, any other power of i can
be derived.

.2
-1
3

4
i 1

.5

.6
—1
7

8
i —1

The results repeat in cycles of 4. You can use the fact that i4 1 to simplify higher powers of i. For
example,
50 2 2
1 —1

When i isused in an expression like 3 + 2i, the expression is called a complex number. We add, subtract,
multiply, and divide complex numbers much like we would algebraic expressions.

EXAMPLE&jfi var9hicbOf (ÉP3i

Just expand and combine like terms.

(2 -3i) = 3+5i -2 + 3i —1 + 8i

Answer (C)

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 THE COLLEGE PANDA

3 EXÄMPLG _Gtventhati isthéproduct 2i)?

A) 18 3i B)22, 3i C) 1843i D) 22431

Expanding,
(4 + i) (5 - 2i) = 20 - 8i + 5i - 212 20 _ 3i + 2 — 22 - 3i

Answer (B) .

EXAMPLP@: Which of the following is equal to

When you're faced with a fraction containing i in the denominator, multiply both the top and the bottom of
the fraction by the conjugate of the denominator. What is the conjugate, you ask? Well, the conjugate of 1 +i
is 1 — i. The conjugate of 5 — 4i is 5 + 4i. To get the conjugate, simply reverse the sign in between.
In this example, we multiply the top and the bottom by the conjugate 1 i.

(1 —i) 2 - 2i + 3i - 3i2 2 i — 31
•2

I—i+i_ p 2

The whole point of this process is to eliminate i from the denominator. The absence of i in the denominator is

a good indicator that things were done correctly. The answer is (D)

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CHAPTER 17 COMPLEX NUMBERS

CHAPTER EXERCISE: Answers for this chapter start on page 291.

A calculator should NOT be used on the

following questions.

If the expression above is equivalent to a + bi,

where a and b are constants, what is the value of
For i —1, which of the following is

equivalent to (5 3i)

B) 12
B) 3+2i
C) 22
D) 34

Given that = V/Z-Ä, which of the following is

i
Which of the following is equal to
equal to i(i + 1)?
+ 2) — -4i)? (Note: i

A) 16 - 5i

C) —4 + Ili
16 Ili

For i — VZÄ, which of the following is

equivalent to 3i(i + 2)
Which of the following is equal to the expression
above? (Note: i -1)

For i — VZ-Ä, which of the following is equal to

i93?

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 THE COLLEGE PANDA

12

Which of the following complex numbers is Which of the following complex numbers is

equivalent to (3 — i)2? (Note: i —1)

equivalent to ? (Note: i —1)

B) 8+6i 3 4
5 5
C) 10 -6i
4
D) 10 i
B) 1 —i
5
5 4
3 3
4
D) 1
3
Which of the following is equal to the expression
above? (Note: i — —1)
A) 14 -7i
B) 14- 23i
C) 26+7i
D) 26 23i

10

Which of the following is equal to — +

(Note: i

11

Which of the following is equal to ? (Note:

—1)

5
C)
4

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 Absolute Value
The absolute value of x, denoted by lxl, is the distance x is from 0. In other words, absolute value makes
everything positive. If it's positive, it stays positive. If it's negative, it becomes positive.

egervaluesofrx €4?

Think of the possible numbers that work and don't forget the negative possibilities. Every integer between —3
and 3 works, a total of integer values.

We could've also solved this problem algebraically. Any absolute value equation like the one above can be
written as

and since x is an integer,

EXAMPLE Of satisfy]$ 5?

Here we go through the same process. The largest possible integer for x is 3 and the smallest is —5. So
—5 x 3, a total of 9 possibilities.

Solving algebraically,
-5 < x+1<5
Subtracting 1,

Éoryhrchof the foÄlowinv valuesof%r 5149i

Trick question. The absolute value of something can never be negative. There is no solution, answer (D)

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 THE COLLEGE PANDA

4: manufacturer of cookies tests the weight of its Cookie paekages to en$Ure

in the product. An acceptable«vackage of cookieS must:weigh 18 ounces as
omes*out of production. theweight of an acceptable cookie package, then which.of the fomowing
Ifew is
inequalities correctly expresses ali possible values ofw?

171>1 'B)Lw 161k 2 D) [w 171<1

In these types of absolute value word problems, start with the midpoint of the desired interval, 17 in this case,
and subtract it from w: lw 171.Think of this as the "distance," or "error," away from the midpoint of the
interval. We don't want this "error" to be greater than 1 since w would then be outside the desired interval. So

our answer is (D) lw 171 < 1.

We can confirm this answer by solving the inequality. Remember that the end result should be 16 < w < 18.
Let's see if our answer gives us that result when we isolate w.

lw 171 < 1

-1 < w 17<1
Adding 17,
16 <w< 18

We have confirmed that (D) is the correct answer.

This is the graph of y x:

Now this is the graph of y lxl:

See how the graph changed? Taking the absolute value of any function makes all the negative y-values become
positive y-values (points in the quadrants Ill and IV are reflected across the x-axis). All the positive y-values
stay where they are. This V-shape is the classic absolute value graph that you should be able to recognize.

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CHAPTER 18 ABSOLUTE VALUE

A table of values is another way to see this absolute value transformation. If f(x) — 2x, then compare f (x)
with If

x —3 -2 —1 0 1 2 3

f(x) —6 -2 o 2 4 6

If(x) I
6 4 2 o 2 4 6

The negative values of f (x) become positive and the positive values of f (x) stay positive.

EXÅMPLE5vWhichofthef0110wingco dbe the gra h ofY$ 12x= 11?

(C)
Y

The entire function is enclosed inan absolute value and since the absolute value of something can never be
negative, y must always be greater than or equal to 0. In other words, the graph must lie on or above the x-axis.
That eliminates (A) and (B). In fact, (A) is the graph of 2x — 1 without the absolute value. To get the answer, we
take all the points with negative y-values in the graph of (A) and reflect them across the x-axis so that they're

positive. The graph we end up with is (C)

One great tactic that's worth mentioning here is narrowing down the answer choices by obtaining points that
are easy to calculate. For example, if we let x = 0, then y I2(O) - II 1. The point (0, 1) must then be on

the graph, eliminating (A) and (B). Letting y 0, we now find that (0.5, 0) must also be on the graph. This

eliminates (D) because (D) has two x-intercepts whereas the graph should only have one.

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 THE COLLEGE PANDA

CHAPTER EXERCISE: Answers for this chapter start on page 292.

A calculator should NOT be used on the

following questions.
If lx — 31 > 10, which of the following could be
the value of lxl?

If f (x) — —2x2 — 3x + 1, what is the value of

A calculator is allowed on the following

questions.

Which of the following expressions is equal to

5 for some value of x? How many different integer values of x satisfy
lx+61 < 3?

B) lx-21-6

D) lx+61-2

If n — 21 10, what is the sum of the two

possible values of n?

C) 12
D) 20

Which of the following could be the equation of

the function graphed in the xy-plane above?

A) y = -lxI-2
xl-2

D) Ix-21

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CHAPTER 18 ABSOLUTE VALUE

If lx 101 = b, where x < 10, then which of the

following is equivalent to b x?

A) —10
B) 10
C) 2b— 10

D) 10-2b
The graph of the function f is shown in the
xy-plane above. Which of the following could be
the graph of the function y If (X) l?

A hot dog factory must ensure that its hot dogs

are between 6 inches and 6— inches in length.

If h is the length of a hot dog from this factory,

x
then which of the following inequalities
correctly expresses the accepted values of h?

1
A) h
B) 4

1
B) h
2
1
x C) h
4

1
D)
4
C)

10

x Rolls of tape must be made to a certain length.

They must contain enough tape to cover
between 400 feet and 410 feet. If I is the length of
a roll of tape that meets this requirement, which
D) of the following inequalities expresses the
possible values of l?

A) II < -4001 10

B) -4051 >
II 5
x
C) 11+4051 < 5

D) -4051 <
II 5

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 THE COLLEGE PANDA

11

If 14x — 41 8 and 15y + 101 = 15, what is the

smallest possible value of xy?

20

B) -15

D) —1

12

If lal < 1, then which of the following must be

true?

1
1.
a
II. a < 1

111. a > 1

A) 111 only
B) I and II only

C) II and Ill only

D) 1, 11, and 111

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 Angles
Exterior Angle Theorem
An exterior angle is formed when any side of a triangle is extended. In the triangle below, xo designates an
exterior angle.

bo

xo

An exterior angle is always equal to the sum of the two angles in the triangle furthest from it. In this case,

EXAMPLE 1:

What:i$ th€yalue Oi Xhn thefigureai)ove

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 THE COLLEGE PANDA

ZDCE must be 800. Now there are a lot of ways to do this, but using the exterior angle theorem is the
fastest:

80 x 3x

80

X 40

Parallel Lines

When two lines are parallel, the following are true:

• Vertical angles are equal (e.g. Zl Z4)

• Alternate interior angles are equal (e.g. Z4 Z5 and Z3 Z6)

• Corresponding angles are equal (e.g. Zl Z5)

• Same side interior angles are supplementary (e.g. Z3 + Z5 1800)

No need to memorize these terms. You just need to know that when two parallel lines are cut by another line,
there are two sets of equal angles:

12 — 13 — Z6 — 17

'EXAMPLE

BF@CE. If LCAE 70%nd LACE—40Swhatistheova1ue ofx?

Here is the fastest way: Z ACE Z ABF 400 because they are corresponding angles (AC cuts parallel lines

BF and CE). Since angle x is an exterior angle to AABF, x = 70 + 40 1100 .

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CHAPTER 19 ANGLES

Polygons
Triangle Quadrilateral Pentagon Hexagon

1800 3600 7200

As you can see from the polygons above, each additional side increases the sum of the interior angles by 1800.
For any polygon, the sum of the interior angles is

180(n — 2) where n is the number of sides

So for an octagon, which has 8 sides, the sum of the interior angles is 180(8 — 2) 180 x 6 — 10800

A regular polygon is one in which all sides and angles are equal. The polygons shown above are regular. If

our octagon were regular, each interior angle would have a measure of 10800 —8 — 1350. •

The 180(n 2) forrnula comes from the fact that any polygon can be split up into several triangles by drawing
lines from any one vertex to the others.

The number of triangles that results from this process is always two less than the number of sides. Count for
yourself! Because each triangle contains 1800, the sum of the angles within a polygon must be 1800 (n — 2),
where n is the number of sides.

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EXAMPLE

IWo sid€OCa as show/irv@he figure abpye. What is the x?

The total number of degrees in a pentagon is 180(5 — 2) 5400. So each interior angle must be 5400 +5 = 1080.
The angles within the triangle formed by the intersecting lines must be 180 — 108 720.

1080
720

1080 720

sox — 180-72 — 72 360 .

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CHAPTER 19 ANGLES

CHAPTER EXERCISE: Answers for this chapter start on page 294.

A calculator should NOT be used on the

following questions.

a bo
m

k0 1

Note: Figure not drawn to scale.

In the figure above, i 50 and k — 140. What is In the figure above, lines and m are parallel.
I

the value of j?
What is the value of a + b + c + d?
A) 60 A) 270
B) 360
B) 70
C) 80 C) 720
D) It cannot be determined from the
D) 90
information given.

600
x

500 400

Note: Figure not drawn to scale. Note: Figure not drawn to scale.

In the figure above, what is the value of y? In the figure above, if x = 40, what is the value
of y?
A) 30
A) 40
B) 40
B) 50
C) 50
C) 80
D) 70
D) 90

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A calculator is allowed on the following

questions.
1

m 450

800
In the figure above, lines and
I m are parallel.
What is x in terms of a and b?

Note: Figure not drawn to scale.

In the figure above, what is the value of x + y?

D) 180 a
A) 125
B) 180
C) 235
D) 280

700
300

450

Note: Figure not drawn to scale.

In the figure above, what is the value of a + b?

A) 80
B) 100 400
600
C) 110
D) 120
Note: Figure not drawn to scale.

In the figure above, what is the value of z?

A) 35
B) 45

C) 55
D) 80

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CHAPTER 19 ANGLES

(x + 40)0 xo

In the figure above, what is the value of x?

A) 60
B) 70

C) 75
D) 80 In the figure above, a rectangle and a
quadrilateral overlap. What is the sum of the
degree measures of the shaded angles?
10
A) 360
B) 540

C) 720
D) 900

800
12

1300

A regular hexagon is shown in the figure above.

What is the value of x?
A) 15
Note: Figure not drawn to scale.
B) 20
In the figure above, what is the value of y? C) 25
A) 100 D) 30
B) 130
C) 140
D) 150

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13 15

1
400
1
450

600
m
m
n

Note: Figure not drawn to scale.

In the figure above, lines I and m are parallel.

Which of the following must be true? Note: Figure not drawn to scale.

I. a 3b
In the figure above, lines l, m, and n are parallel.
What is the value of a + b?
111. b 45

A) Ill only

B) I and II only

C) II and Ill only

D) 1, 11, and 111

14

700

1000

Note: Figure not drawn to scale.

In the figure above, what is the value of x + y?

A) 10
B) 20
C) 30
D) 50

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 20 Triangles
Imagine you are trying to make a triangle with sticks of different sizes.

10

1
1

Can you make a triangle with any 3 sticks? No. In this case, the two shorter pieces don't connect.

10

If you had sticks of size 5, 5, and 10, they would connect but only by just enough to make a straight line.

5 5

10

So to make a triangle, the lengths of any two sticks must add up to be greater than the third. To say it more
mathematically,

For any triangle, the sum of any two sides must be greater than the third.

a+b>c
b

If the sum of two sides turns out to be equal to the third, it's enough to make a line, but NOT a triangle.

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m 44BC»ÄBhaS .Äengthcof.3 andjC has atléngthof4, mtebervaltie$ arc

possible for the length of

The golden rule of any geometry problem is to draw a picture:

Let the length of AC be x. Based on the rule, we can

come up with three equations:

3+4 > x

3 x

which simplify to

Now if x > 1, then it's always going to be greater than —1. In other words, only the first and second matter.
Therefore, 1 < x < 7, and there are possible integer values of x.

Isosceles & Equilateral Triangles

An isosceles triangle is one that has two sides of equal length. The angles opposite those sides are equal.

c
Because AB AC, ZC — LB.

In an equilateral triangle, all sides have the same length. Because equal sides imply equal angles, the angles
are all 600.

600

600 600

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CHAPTER 20 TRIANGLES

ÉXÅMPLEä:han asoscelesånangléy one of angles has ineasttre of What is the degree measure
of the greatest possible angle the triangle?

An isosceles triangle has not only two equal sides but also two equal angles. There are two possibilities for an
isosceles triangle with an angle of 500. Another angle could be 500, making a 50 — 50 80 triangle, or the other

two angles could be equal, making a 50 65 — 65 triangle. Given these two possibilities,@fl is the greatest
possible angle in the triangle.

EXAMPLG 3:

ove; the tnangleABC1.s equilateral What is tne value of

Solution 1: There are 3 smaller triangles within the equilateral one. Each of these triangles has a total degree
measure of 1800, for a combined total of 1800 x 3 5400. We need to subtract out ZACB to get what we want.

Because triangle ABC is equilateral, ZACB is 600. So 5400 — 600 4800

Solution 2: Because AABC is equilateral, both j and o are 600. Because k and I form a straight line, they add
up to 1800. Because m and n also form a straight line, they also add up to 1800. Adding up all our values, we
get 600 —I— 1800 + 1800 + 600 4800 .

Right Triangles
Right triangles are made up of two legs and the hypotenuse (the side opposite the right angle).

Every right triangle obeys the pythagorean theorem: a2 + b2 c2, where a and b are the lengths of the legs and
c is the length of the hypotenuse.

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EXAMPLE

20

has of length 20. Ifthe base •of åie rectang!eis -twice as long as the heighi,
whåt IS •tKe
eat O, O:

The diagonal of any rectangle forms two right triangles. Let the height be x and the base be 2x. Using the
pythagorean theorem,
x2 + (2x)2 = 202
12 + 4x — 400
2

x — 80

x 80 4v/S

If you take the SAT enough times, what you'll find is that certain right triangles come up repeatedly. For
example, the 3 — 4 — 5 triangle:

5
3

A set of three whole numbers that satisfy the pythagorean theorem is called a pythagorean triple. Though not
necessary, it'll save you quite a bit of time and improve your accuracy if you learn to recognize the common
triples that show up:

5, 12, 13

7, 24, 25

8, 15, 17

Note that the 6 8 10 triangle is just a multiple of the 3 —4 5 triangle.

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Special Right Triangles

You will have to memorize two special right triangle relationships. The first is the 450 — 450 900:

450

450

The best way to think about this triangle is that it's isosceles—the two legs are equal. We let their lengths be x.
The hypotenuse, which is always the biggest side in a right triangle, turns out to be times x.

We can prove this relationship using the pythagorean theorem, where h is the hypotenuse.

2x 2

I show you these proofs not because they will be tested on the SAT, but because they illustrate problem-solving
concepts that you may have to use on certain SAT questions.
The second is the 300 — 600 — 900:

600

2x
x

300

Because 300 is the smallest angle, the side opposite from it is the shortest. Let that side be x. The hypotenuse,
the largest side, turns out to be twice x, and the side opposite 600 turns out to be v/ä times x.

One common mistake students make is to think that because 600 is twice 300 , the side opposite 600 must be
twice as big as the side opposite 300. That relationship is NOT true. You cannot extrapolate the ratio of the
sides from the ratio of the angles Yes, the side opposite 600 is bigger than the side opposite 300, but it isn't

twice as long.

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We can prove the 30 — 60 — 90 relationship by using an equilateral triangle. Let each side be 2x (we could use
x but you'll see why 2x makes things easier in a bit):

600

600 600

Drawing a line down the middle from B to AC creates two 30 — 60 — 90 triangles. Because an equilateral
triangle is symmetrical, AD is half of 2x, or just x. That's why 2x was used—it avoids any fractions.

300

600 600

x c

To find B D, we use the pythagorean theorem:

AD2 + BD2 AB2

x2 + BD2 (2x)2

BD2 (2x)2 — x2

BD2 = 4x2 — x2
BD2 3x2

Triangle ABD is proof of the 30 — 60 90 relationship.

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CHAPTER 20 TRIANGLES

EXAMPLE 5:

450

450

What is the area of AÄCB shown above?

B)2Vi

Using the 45 — 45 90 triangle relationship, AC (the hypotenuse is times greater than each

leg). The area is then 1 (16)

kVi
Answer (C) .

EXAMPLÉ 6:

10

300

In the figure above, AD DC, LB 300, and AB 10. What is the ratio of AC to CB?

B) D) 3

Because AD DC, A ADC is not only isosceles but also a 45 — 45 — 90 triangle. AADB is a 30 — 60 — 90
triangle with a hypotenuse of 10. Using the 30 — 60 — 90 relationship, AD is half the hypotenuse, 5, and
DB 5Vj. Using 45-45-90 relationship, = 5vé, DC 5, and CB DB DC — 5V3 5.

5v/i
1

Answer

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Similar Triangles
When two triangles have the same angle measures, their sides are proportional:

Because DE is AC in the figure above, ZBED is equal to ZBCA. That makes ADBE similar to
parallel to
AABC. In other words, ADB E is just a smaller version of AABC. If we draw the two triangles separately and
give the sides some arbitrary lengths, we can see this more clearly.

5
10 4
8

6 c

The sides of the big triangle are twice as long as the sides of the smaller one. Even if the lengths change, the
ratios will remain the same:
AB AC BC
BD DE BE

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CHAPTER 20 TRIANGLES

EXÄMPLE 7:

c
Note: Figure notdrawn to scale,

IntÄABCabove, DEis payalleito AD 4UDB 43, anCDE 6. What is length ofÄC?

PART p What is the of the arte of ABDF t? ABAC?

B)
25

Part 1 Solution: Because DE and AC are parallel, Z BDE is equal to Z BAC and Z BED is equal to ZBCA.
Therefore, ABDE and ABAC are similar. Setting up our ratios,
DE
AC
3 6
5

Cross multiplying,
3AC 30

10

Part 2 Solution: When two triangles are similar, the ratio of their areas is equal to the square of the ratio of

their sides. The ratio of the sides is 3 : 5. Squaring that ratio, we get the ratio of the areas, 9 : 25. Answer (D)

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Radians
A radian is simply another unit used to measure angles. Just as we have feet and meters, pounds and
kilograms, we have degrees and radians,
radians 1800

If you've never used radians before, don't be put Off by the After all. s just a number. We could've
written
3.14 radians z 1800
instead, but everything is typically expressed in terrns of when we*re working with radians. Furthermore,
3.14 is only an approximation. So, given the conversion factor above, how would we convert 450 to radians?

radians
450 x. — radians
1800 4

Notice that the degree units (represented by the little circles) cancel out just as they should in any conversion

problem. Now how would we convert to degrees? Flip the conversion factor.

37T 1800
radians x
2 7t radians

You might be wondering why we even need radians. Why not just stick with degrees? Is this another difference
between the US. and the rest of the world, like it is with feet and meters? Nope. As we'll see in the chapter on
circles, some calculations are much easier when angles are expressed in radians.

EXAMPLE
y

In the xy-plane above, line m passes through one origin and has a slope of If point A lies line m
and point B lies on x-axis as shown, what is tneasure. in radians, of angle AOB?

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CHAPTER 20 TRIANGLES

We can draw a line down from A to the x-axis to make a right triangle. Because the slope is v/6, the ratio of the
height of this triangle to its base is always to 1 (rise over run).

x
1

This right triangle should look familiar to you. It's the 30 60 — 90 triangle. Angle AOB is opposite the v/ä, so
its measure is 600. Converting that to radians,

600 x
1800 3

Answer (D) .

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CHAPTER EXERCISE: Answers for this chapter start on page 296.

A calculator should NOT be used on the

following questions.

x
The lengths of the sides of a right triangle are x, 6
x — 2, and x + 5. Which of the following
equations could be used to find x?

A square of side length 6 is shown in the figure

above. What is the value of x?
2

D) 6N/6

10

600 600

c
3
Note: Figure not drawn to scale.
D
In ABDC above, what is the length of DC? 6
c

In the figure above, AB II CD. What is the length

of AB?

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CHAPTER 20 TRIANGLES

The lengths of two sides of a triangle are 3 and

13. What is one possible length of the third side?
5 5

What is the area of isosceles triangle MNO

above?

Two angles of a triangle have the same measure.

If two sides have lengths 15 and 20, what is the
greatest possible value of the perimeter of the

triangle?

Which of the following sets of the three numbers

could be the side lengths of a triangle?

1. 6, 14, 7 In the figure above, an equilateral triangle sits on

11. 5, 5, 12 top of a square. If the square has an area of 4,
111. 4, 8, 11 what is the area of the equilateral triangle?

A) I only
B) Ill only
C) II and Ill only 2

and 111 3
D) 1, 11,

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10 12

2
1

Note: Figure not drawn to scale.

B H 10 c
In the figure above, the base of a cone has a
Note: Figure not drawn to scale.
radius of 6. The cone is sliced horizontally so
that the top piece is a smaller cone with a height
AB is parallel to GH and DF
In the figure above,
of and a base radius of 2. What is the height of
1
is parallel to DE — 1, EH — 3, EG 2, and
BC. If
the bottom piece?
HC — 10, what is the length of AD?

The lengths of three sides of a triangle are x, y, A calculator is allowed on the following
and z, where x y z. If x, y, and z are integers questions.
and the perimeter of the triangle is 10, what is
13
the greatest possible value of z?

How many radians are in 2250?

37T

4
77T

6
571

4
37T
2

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CHAPTER 20 TRIANGLES

14 16

9
6 5

B 9

Triangle ABC above is similar to triangle DEF.

What is the perimeter of triangle DEF?
A) 20
B) 26.8
In A ABC above, ZCDE -z: 900 and ZA 900.
C) 30 AB 9 and AC = 12. If DE = 6, what is the
D) 36.2 length of C E?

15

The lengths of two sides of a triangle are 8 and D) 10

20. The third side has a length of p. How many

positive integers are possible values of p?

17
A) 14
B) 15
In isosceles triangle ABC, BC is the shortest side.
C) 16 If the degree measure of ZA is a multiple of 10,
D) 17 what is the smallest possible measure of LB?
A) 750
B) 700

C) 650
D) 600

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18 20

z
20
x
12
What is the value of x in the triangle above?

x 15

Two poles represented by X W and YZ above are

15 feet apart. One is 20 feet tall and the other is
12 feet tall. A rope joins the top of one pole to the
top of the other. What is the length of the rope?

A) 12 21

17
B x c
C) 18
D) 19

19

A z

In the figure above,ABCD is a square of side

25 length 3. If AWAZ CX — CY = 1, what is
24
the perimeter of rectangle WXYZ?

35 B) 4v6
What is the perimeter of the trapezoid above?

A) 100
B) 108
C) 112
D) 116

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CHAPTER 20 TRIANGLES

22 23

B(m, n)

B(-2, -3)

Two parallel lines are shown in the xy-plane

above. If AB 15 and point B has coordinates
Points A, B, and C form a triangle in the
xy-plane shown above. What is the measure, in (m, n), what is the value of n?
radians, of angle BAC? 6

12
4

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24 25

600

450

450

What is the length of DB in the figure above?

In the figure above, equilateral triangle ABC is
inscribed in circle D. What is the measure, in
radians, of angle ADB?
27T
B)

37T

C)
47T

57T

26

In the figure above, circle O is inscribed in the

square ABCD. If BD = 2, what is the area of the
circle?

37T

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CHAPTER 20 TRIANGLES

27 29

A triangle has one side of length 5 and another The sides with positive area have
of a triangle
side of length 11. Which of the following could lengths 5, 7,The sides of a second triangle
and a.

be the perimeter of the triangle? with positive area have lengths 5, 7, and b.
1. 20
Which of the following is NOT a possible value
of la b]?
11. 26
111. 30

A) II only

B) I and II only
C) II and Ill only D) 10

D) 1, 11, and 111

30

28
c

In the figure above, equilateral triangle BEC is

contained within square ABCD. What is the
c degree measure of ZBEC?
Equilateral triangle DEF is inscribed in A) 600
equilateral triangle ABC such that ED -L AC. B) 1000
What is the ratio of the area of ADEF to the area C) 1200
of ABC?
D) 1500

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31 33

In the figure above, a semicircle sits on top of a

square of side 6. A is at the top of the
Point
semicircle. What is the length of A B?
In the xy-plane above, angle 0 is formed by the

A) 3VS x-axis and the line segment shown. What is the

measure, in radians, of angle 0?

57T

D) 3 10 3
77T

4
32
97t

5
In AABC, AB — BC 6 and ZABC 1200.
117T
What is the area of AABC? D)
6
A) 2v6

O 4 c

In the figure above, square DBCE has a side

length of 3. If OE 4, what is the length of AD?

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CHAPTER 20 TRIANGLES

35

12

Square ABCD above has a side length of 12. If

BF 4, what is the length of BE?

c) 3v"ä
D) 4Vä

36

D 3

In the figure above, AB = 12, AC = 13, and

DE -z: 3. What is the length of AE?

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 Circles
Circle Facts You Should Knovy•e

Areaof a circle:.Ttr2

Circumference Of a circle: 21tr

Arc Length; 30 x 2m OR 0t if is in radianS

Area of a Sector:z—— X nr =r20 ifOis in radians

360

Central angles have the same measure as the arcs that they "carve out."

Many student$ confuse arc length with Atc measUre'The atc•length is;the actual distance One Would travél
along the circle from A to B. Arc meaSureiSthe number Of degrees one turns through from to Yo
canihin&of it as a rotation along the circlefrom A to B, full rotation is 360%

inscribed angles are half the measure of the arcsthat they icarve oue.jj

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CHAPTER 21 CIRCLES

Angles inscribed in @ semicircle are always 900. This is just an extension Of the previous fact. An angle
inscribed in a semicircle carves out half a circle, or 1800 , which means the angle itself is half that, or

.A 'fadius drawn to a line tangent to the circle is perpendicular to that line:

General equation of a Circle in the xy-plane:

where (h,'k) IS the center of the circle and r isits radius.

EXAMPLE 1;

TO the figure above/the outercircle'sradius is twice as long as the inner circle's. What is the ratio of the
area Of the shaded region to theatea of the unshaded region?

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Let the radius Of the inner circle be Then the radius Of the outer circle is 2r.

Area of inner circlet yrr2

Area of outer circle:
Area of shaded region: — mr2 4nr2 — 3mr2

Shaded 3Ttr2
Unshaded

The answer is (D)

EXAMPLE Z

c 2

What is the area of shaded region in the figure above?

A)å-Vi c)E2

To get the shaded area, we must subtract the area of the triangle from the area of the sectore

450
Area of sector: —nr2 — —n(22) —

Area of triangle: Draw the height from point A to base CB. This makes a 45 45 — 90 triangle. Because AC is
2
also a radius* its length is 2. Using the 45 — 45 — triangle relationship, the height is then

Area — bh

Area of shaded region =

2

The answer is (D)

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CHAPTER 21 CIRCLES

EXAMPLE 3:

1209M

Agirgle With a diameter of 10 is shown in the figure above. If ZAQB 1200 , what is the length of minor
arc

257t 207T 107T 57t

C)

1200 107T
(27Tr) — —(27T X 5) —
3600

The answer is (C)

EXÅMPLE 4:

In the figure above, ZACB is inscribed m circle O, What is the measure of angle.ÄCB?
A) 150 B) 300 C)450 D)600

The measure of minor arc AB is the same as the measure of central angle ZAOB, 900. Inscribed angle ACB is
half of that, 450. Answer (C)

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;EXAMPLEB:,
y?A42y...31
ascircie thexyolåne ISViven above.ÄMhat are the of the-center/f the ctrclei

To get the equation of the circle in the standard form (x + (y r2, we have to complete the square

twice, once for the x's and once for the y's. If you don't know how to complete the square, you should review
the quadratics chapter, which contains many examples of how to do it. Starting with x,

x — —4 + Y2 +2y = 31

Then y,
1=31
36

From the standard form, we can see that the center is at (2, —1) and the radius is 6. Answer (D)

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CHAPTER EXERCISE: Answers for this chapter start on page 302.

A calculator is allowed on the following

questions.

The circle above has area 367T and is divided into

8 congruent regions. What is the perimeter of
one of these regions?
c
A) 6 I .57T

In the figure above, the square ABCD is B) 6427T

inscribed in a circle. If the radius of the circle is r,
C) 12 I .57T
what is the length of arc APD in terms of r?
D) 12+27T

4
7tr

2
Which of the following is an equation of a circle
2 in the xy-plane with center (—2, 0) and an area
of 49m?
4
A) 7

B) 7
2
49
49

In the figure above, three congruent circles are

tangent to each other and have centers that lie on
the diameter of a larger circle. If the area of each
of these small circles is 97T, what is the area of
the large circle?

A) 367T

B) 497T

C) 647T
D) 817T

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300
c
c

Note: Figure not drawn to scale.

In the figure above, equilateral triangle ABC is
In the figure above, ZAC B is inscribed in a inscribed in circle D. If the area of circle D is
3671, what is the length of minor arc AB?
circle. The length of minor arc AB is what
fraction of the circumference of the circle? A) 27T

1 B) 37T
3 C) 47T
1
D) 67T
B)
4
1

6
1

12

c
6

c In the figure above, circle C has a radius of 6. If

the area of the shaded sector is 1071, what is the

measure, in radians, of angle ACB?
27T

In the figure above, AC is a diameter of the circle 5

and the length of AB is 1. If the radius of the 47t
circle is 1, what is the measure, in degrees, of 9
ZBAC? 57T

9
57T

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11

In the figure above, a circle has center C and

The base of a right circular cylinder shown
radius 5. If the measure of central angle ACB is above has a radius of 4. The height is 5. What is
between — and radians, what is one possible the surface area of the cylinder?

integer value of the length of minor arc AB? A) 4071

C) 727T
D) 8171

12

10

In the figure above, circle P and each

circle II
have a radius of 3 and are tangent to each other.
In the figure above, four circles, each with radius
4, are tangent to each other. What is the area of
If APHLI is equilateral, what is the area of the

the shaded region?

shaded region?

A) 16 47T
A) 107T

B) 127T
B) 64
87T C) 147t
C) 64
D) 157T
D) 64 167T

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13

The equation of a circle in the xy-plane is given

above. Which of the following must be true?
I. The center of the circle is at (2, 4)
II. The circle is tangent to the x-axis.
Ill. The circle is tangent to the y-axis.

A) 11 only

B) 111 only

C) I and II only
Note: Figure not drawn to scale.
D) 1, 11, and 111

If the area of the shaded region in the figure

above is 247T and the radius of circle O is 6, what
is the value of x?

A) 15
B) 30

C) 45
D) 60

14

In the figure above, circle A is tangent to circle B

at point D. If the circles each have a radius of 4
and AC is tangent to circle B at point C, what is
the area of triangle ABC?

D) 16

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 Trigonometry
22
We illustrate the three trigonometric functions you need to know with a 5 — 12 — 13 right triangle.

13
5

x
12

opposite 5 adjacent 12 opposite 5

sin x — cos x tan x
hypotenuse 13 hypotenuse 13 adjacent 12

It's important to see these trigonometric functions as if they were just ordinary numbers. After all, they're just

ratios. For example, sin 300 is always equal to . Why? Because all right triangles with a 300 angle are similar.

The ratios of the sides stay the same.

4
2

2
1

300 300

Many students over-complicate trigonometry because they treat sin x, cos x, and tan x differently than regular
numbers. Perhaps because of the notation, students sometimes make mistakes like the following:

sin 2x
sin 2
x

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 THE COLLEGE PANDA

The above is not possible because sin2x is one "entity" You cannot separate sin and 2x and treat them
independently just like you can't separate f (x) info f and x.

The definitions of sine, cosine, and tangent are best inemo,rized through the acronym SOH-CAH-TOA;S
for sine (opposite over hypotenuse), C for cosine (adjacent over hypotenuse), and T for tangent (opposite
over adjacent)

Aside from the definitions, you should also memorize the following very important identity:

sinx ccxs(900

The reverse is also true.

COSX sin(900 X)

Expressed in •

sin* eos and cos x = sin

Now, the sign of each of the trig functions depends on the quadrant in which the angle terrninates.

11 1

111

• Sine. ccsine.and tangent are all positive in the first quadrant.

• y sine is positive in the second quadrant.

• Only tangent is positive in tlw third quadrant.

Only cxsine is positive in fourth quadrant.

These are best memorized through acronym ASTC (All Studcmts Take Calculus). All the functions are
positive in the first quadrant, only sine is positive in the secondi and so on.

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CHAPTER 22 TRIGONOMETRY

To find theyalueofå tng ftincåion for amangle withOui a Calculator,

I. Determine-what the sigvofthe result shotildbe (poSftive or negafiveb

2. Add or subtract multiplesof909 from«theangle Cor radians) until you get an angle in the first

quadrants

45Z'4S.90 or 30—60 90speciå1 righttriangrestogetthe value; ThéSATwon'taskyoti,

to catcylate tng valuesior anglés thaQ arenit in these spectal righttrianglesunless you re able to use
youVea1Cuiatog
4 ÄåaYsure.your result has the correct stgnofrom step one

Let's do a couple simple examples.

1. What is the value of sin 3300?

Since 3300 is in the fourth quadrant and sine is negative in the fourth quadrant, the result should be
negative. Subtracting 900 from 3300 until we get an angle in the first quadrant, we end up with 330
270 600. Using the 30—60 90 triangle,

600
2
1

300

sin 600
opp
hyp 2

Since the result should be negative, sin 3300

2
2. What is the value of cos 1350?

Since 1350 is in the second quadrant and cosine is negative in the second quadrant, the result should be
negative. Subtracting 900 from 1350, we get 450. Using the 45 — 45 — 90 triangle,

450

450

adj 1
cos 450
hyp 2

Since the result should be negative, cos 1350

2

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 THE COLLEGE PANDA

3. What is the value of tan 2100?

Since 2100 is in the third quadrant and tangent is positive in the third quadrant, the result should be
positive. Subtracting 900 from 2100 until we get an angle in the first quadrant, we end up with 210
180 = 300. Using the 30 — 60 — 90 triangle shown earlier,

tan 300
3

Since the result should be positive, tan 2100

3

Finally, you sthOuld memorize the following valueSfo±00 and 90

1 •cos
fann.•u tanA)Ø un ed

4
2
—tan —undefined

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CHAPTER 22 TRIGONOMETRY

CHAPTER EXERCISE: Answers for this chapter start on page 304.

A calculator should NOT be used on the

following questions.
In right triangle ABC, the measure of ZC is 900
5
and AB = 30. If cos A what is the length of
6'
If cos 400 a, what is sin 500 in terms of a?

C) 90 —a
D) av/ä

In a right triangle, one angle measures xo such If tan x — m, what is sin x in terms of m?
that tan xo = 0.75. What is the value of cos xo? 1

m2 + 1

1
B)
1 — m2
m
C)
m2 + 1

m
D)
1 — m2

sin 0 + cos(90 0) + cos 0 + sin(90 — 0)

For any angle 0, which of the following is

equivalent to the expression above?

B) 2 sin O

C) 2coso 5

D) 2(sin0 + cos 0)
c

Given that AB = 4 and tan B in the right

triangle above, what is the value of

sin B + cos B?

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 THE COLLEGE PANDA

cos 32 — sin 5m 12
A calculator is allowed on the following
questions. In the equation above, the angle measures are in
degrees. If 00 <m< 900, what is the value of m?

c 10

If sin x 0.25, what is the length of BC in the

Y
triangle above?

C(12, -3)

Right triangle ABC is shown in the xy-plane

above. What is the value of cos C?
8
17

In the figure above, right triangle ABC is similar 8

to right triangle MNO, with vertices A, B, and C 15
corresponding to vertices M, N, and O, 13
respectively. If tan B = 2.4, what is the value of 15
cos N?
15
17

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CHAPTER 22 TRIGONOMETRY

11

In the figure above, AC is a diameter of the

circle. If AC — 1, which of the following gives
the area of triangle ABC in terms of 0?
0
2
tan 0
B)
2

C) 2sine
sin 0 cos 0
D)
2

12

Given that sin 9 — cos 0 0, where e is the

radian measure of an angle, which of the

following could be true?

11.
2
37T
111.
2

A) I only

B) II only

C) 1 and 111 only

D) 1, 11, and 111

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 Reading Data
The SAT loves to test your ability to read graphs and charts. Fortunately, these are typically the easiest
questions because they never involve too much math. Most of them just test you on simple arithmetic with the
extra step of having to interpret a graph. Practice away!

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CHAPTER 23 READING DATA

CHAPTER EXERCISE: Answers for this chapter start on page 306.

A calculator is allowed on the following

questions.
Voter Turnout in
Congressional and Presidential Elections

Commute Times 80
90 75 Congressional Election
Presidential Election
c 70
75
o 65

E 60 60
e 55
45
50

30 8 45
40
15 35
30
oo
15 30 45 60 75 90 O
o O
O o o o o
Commute Time to Work 01

Year
For four work days, Alex plotted the commute
time to work and the commute time from work in The graph above shows the voter turnout for
the grid above. For which of the four days was each year a congressional election or a
the total commute time to and from work the presidential election was held. In which two
greatest? year period was the difference in voter turnout
between the congressional election and the
presidential election the smallest?

A) 1996 to 1998
B) 2000 to 2002
C) 2004 to 2006
D) 2008 to 2010

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 THE COLLEGE PANDA

Ice Cream Sales Average Precipitation in Kathmandu

1.75 400
1.70 350
300
1.65
o 250
1.60
200
1.55 150
o 100
1.50
50
1.45

1.40

1.35
Month

The line graph above shows the monthly

precipitation in Kathmandu last year. According
to the graph, the total precipitation in September
was what percentage of the total precipitation in
According to the line graph above, ice cream June?
sales were highest both in 2013 and in 2014
A) 400/0
during which three month period?

A) January to March
B) April to June D) 75%
C) July to September
D) October to December

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CHAPTER 23 READING DATA

Population in 2010 Birth Rate

Actual a South Korea

Estimated 40
Japan
San Diego

30
Chicago

20
Los Angeles

1 2 3 4 10
2006 2008 2010 2012 2014
Population (millions) Year

Researchers created the graph above to compare Based on the graph, which of the following best
their population estimates with the actual describes the general trend in birth rates in
populations of different cities in 2010. For which South Korea and Japan from 2006 to 2014?
of the cities did the researchers underestimate
A) Each year, birth rates decreased in both
the population?
South Korea and Japan.
I. San Diego Each year, birth rates increased in both
B)
Chicago South Korea and Japan.
111. Los Angeles
C) Each year, birth rates increased in South
A) I only Korea but decreased in Japan.
B) I and II only
D) Each year, birth rates decreased in South
C) II and Ill only Korea but increased in Japan.
D) 1, 11, and 111

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 THE COLLEGE PANDA

30
10
28
26
24
8 22
20
7
18
Q.) 16
14

10 20 30 40 50 60 70 12

Age (years)

Hours since 9:00 A.M.

In a certain study, researchers created the
scatterplot above to summarize the ages of the Starting at 9:00 A.M. each day, Musa picks up
participants and the number of hours of sleep packages at various locations until his trailer
they required each night. Which of the following truck reaches its maximum capacity. He then
is the closest to the age, in years, of the
he picked up that
delivers all the packages that
participant who required the least amount of day. The graph above shows the weight of his
sleep each night? truck at different points during the day. What is
A) 35 the maximum weight Musa's truck can hold, in
tons ?
B) 40

C) 55 A) 14

D) 60 B) 16

C) 24
D) 30

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CHAPTER 23 READING DATA

11

Annual Salt Production in the U.S. Video Game Console Sales in 2015

o
40
250

30 200
o
150
0 20
o
100
o
10
Z'
50
o

2009 2010 2011 2012 2013 2014 2015

B c D E
Year
Console
Based on the graph above, for which of the
The graph above shows the number of units sold
following two consecutive years was the percent in 2015 for five different video game consoles.
increase in U.S. annual salt production the same
The prices of consoles A, B, C, D, and E are $100,
as the percent decrease from 2010 to 2011?
$150, $200, $250, and $300, respectively. Which
A) 2009 to 2010 of the five consoles generated the most total

B) 2012 to 2013 revenue?

C) 2013 to 2014
D) 2014 to 2015

10

400

300
o
200

100

cow Wolf Goat Cat Pig

Animal

According to the graph above, the average mass

of a wolf's brain is what fraction of the average
mass of a pig's brain?

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 THE COLLEGE PANDA

12 13

State Health Care Spending in 2013

20 e Company X
Company Y
15
0 15
o
o
o
o 10
10 o

O 5 5

0
1 2 3 4
AL AZ
Quarter State

The graph above shows the profit of Company X The graph above shows the health care spending
and Company Y in each quarter of last year. In of four different states, Alabama (AL), AK
which quarter was Company X's profit twice (Alaska), AZ (Arizona), and AR (Arkansas) in
Company Y 's? 2013. Based on the graph, which state had the
highest combined hospital care and prescription
drug spending in 2013?

A) Alabama
B) Alaska

C) Arizona
D) Arkansas

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CHAPTER 23 READING DATA

15

190
10
180

170

160

150

140
4
130
unne
120

110

Time (hours)
345678
Time (Hours after 8:00 A.M.)

Jeremy works at a call center. The graph above Greg eats breakfast at 8:00 A.M. and lunch at
shows the average number of calls he answered 12:00 P.M. During each meal, doctors record his
per hour during his 7-hour work shift. What is glucose levels in the graph above until they are
the total number of calls he answered during his
able to calculate Greg's glucose recovery time, the
shift?
time it takes for the body's glucose levels to
return to their recorded value at the start of the
meal. According to the graph, by how many
hours is Greg's glucose recovery time after
dinner greater than his glucose recovery time
after lunch?

A) 1.5

D) 5.5

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 THE COLLEGE PANDA

Car X

50
45
40
35
30
25
20
— 15

0 10 20 30 40 50 60 70

Speed (miles per hour)

The graph above shows the gas mileage for Car

X at different speeds. Based on the graph, how
many gallons of gas are needed to drive Car X
for 5 hours at a constant speed of 30 miles per
hour?

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 24
Probability
Generally speaking, probability can be defined as

number of target outcomes

number of total possible outcomes

Nearly all probability questions on the SAT will involve tables of data. So for the purposes of the SAT,
probability can more narrowly be defined as

number in target group

number in group under consideration

EXAMPLE 1:
Beef -Chicken

First-Class 18
Coach 138

above Of passengers flight, If g first •J@$S

+assenger IS chosen at random from this flight, whatis théoprobabnity thåt the passengerkh(Fen prefers
beef?

The number of first class passengers is 18 + 27 = 45. This is the group under consideration. The number of
first class passengers who prefer beef is 18. This is the target group.

number in target group 18 2

number in group under consideration 45 5

Answer (B) .

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 THE COLLEGE PANDA

2: The manager Of a large asegibly line uses tkte below to —trackp/Uwnumber 01

•vehicles. Chat •dåy,

cars Trucks
Shift 173 126 299
Secondshift i043 :025
-Third shift 165 109 274
520 *898

If a vehtclé is selecteet at of the day, Gllowjng isclosesuO bieprobabili!y

that the a c.r„produced during the first Shift or a Cråékproduéeå during the third

B) 0314 00-952 D)O.42!.

In this question, the groupunder consideration includes all the vehicles, a total of 898 at the end of the day.
The target group includes cars produced during the first shift and trucks produced during the third shift, a
total of 173 + 109 = 282 vehicles.

number in target group

number in group under consideration
—
282
898
0.314

Answer (B) .

CHAPTER EXERCISE: Answers for this chapter start on page 308.

A calculator is allowed on the following questions.

Violation Type

Speeding Stop sign Parking Total

Truck 68 39 17 124
Car 83 51 26 160
Total 151 90 43 284

A district police deparment records driving violations by type and vehicle in the table above. According
to the record, which of the following is closest to the proportion of stop sign violations committed by truck
drivers?

A) 0.137
B) 0.315

C) 0.433
D) 0.567

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CHAPTER 24 PROBABILITY

Questions 2-3 refer to the following information.

The table below shows the number of workers in California with at least one year of experience in five different
construction-related occupations.

Years of Experience
1 2 3 4 5+ Total

Painter 22,491 26,973 29,086 33,861 37,061 149,472

Roofer 23,908 27,634 30,932 34,146 39,718 156,338
Welder 27,062 29,812 32,784 36,902 42,680 169,240
Plumber 28,637 33,119 36,670 40,083 45,376 183,885

Carpenter 24,396 28,806 34,867 37,418 43,922 169,409

Total 126,494 146,344 164,339 182,410 208,757 828,344

Based on the table, if a plumber in California is chosen at random, which of the following is closest to the
probability that the plumber has at least four years of experience?
A) 0.10

B) 0.22

C) 0.25
D) 0.46

If a worker with at least four years of experience is chosen at random from those included in the table, which
of the following is closest to the probability that the person is a plumber?

A) 0.10

B) 0.22

C) 0.25
D) 0.46

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 THE COLLEGE PANDA

Color Red Blue Black Silver

Percent 33% 140/0

A car manufacturer produces cars in red, blue, black, white, and silver. The incomplete table above shows
the percentage of cars it produces in each color. If a car from the manufacturer is chosen at random, what is
the probability that the car's color is red or silver?

A) 23%
B) 330/0

D) 43%

Won Lost Total

Underdog 10 35 45
Favorite 25 5 30
Total 40 75

The table above shows the results of a baseball team, categorized by whether the team was considered the
game or the underdog (expected
favorite (expected to win) in the to lose). What fraction of the games in
which the team was considered the underdog did the team win?

2
7
2
9
2
15

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CHAPTER 24 PROBABILITY

Weekl Week2 Week3 Week4 Total

Box springs 35 40 55
Mattresses 47 61 68 198
Total 82 101 88 77 348

A store manager summarizes the number of box spring and mattress units sold over four weeks at a bedding
store in the incomplete table above. Weeks 2 and 3 accounted for what fraction of all box spring units sold?
2
15
4
15
2
5
4
5

Country Gold Silver Bronze Total

USA 29 29 104

China 38 23 88
Russia 24 26 32 82
Great Britain 29 17 19 65

Germany 11 19 14 44
Total 148 118 117 383

The table above shows the distribution of medals awarded at the 2012 London Summer Olympics. If an
Olympic medalist is to be chosen at random from one of the countries in the table, which country gives the
highest probability of selecting a Bronze medalist?

A) USA
B) Russia
C) Great Britain

D) Germany

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 THE COLLEGE PANDA

Number of Fish Species

Cartilaginous Bony
Philippines 400 800
New Caledonia 300 1,200

All fish can be categorized as either cartilaginous or bony. The data in the table above were produced by
biologists studying the fish species in the Philippines and New Caledonia. Assuming that each fish species
has an equal chance of being caught, the probability of catching a cartilaginous fish in the Philippines is how
much greater than the probability of catching one in New Caledonia?
2
15
1

4
3
10
1

Lightning-caused fires Human-caused fires Total

East Africa 65
South Africa 30
Total 135 220

The incomplete table above summarizes the number of wildfires that occurred in two regions of Africa in
2014 by cause. Based on the table, what fraction of all wildfires in East Africa in 2014 were human-caused?

11
24
13
27
13
24
11
15

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CHAPTER 24 PROBABILITY

Defective Not defective Total

Assembly Line A 300 5,700 6,000

Assembly Line B 500 3,500 4,000

Total 800 9,200 10,000

A manufacturer uses two assembly lines to produce air conditioners. The results of each assembly line's
quality control are shown in the table above. If a refrigerator from the manufacturer turns out to be defective,
what is the probability that the refrigerator was produced by Assembly Line A?

B) 37.5%
C) 600/0

D) 62.5%

11

Type of Residence
Family members Apartment Duplex Single residence Total
1 10 22 3 35
2 20 12 13 45
3 8 8 12 28
4 or more 8 4 18 30
Total 46 46 46 138

The table above summarizes the distribution of living situations for residences in a neighborhood. If a
duplex in the neighborhood is to be inspected at random, what is the probability that the residence is
occupied by no more than 2 family members?

2
23
6
23
17
69

23

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 THE COLLEGE PANDA

12

Percent of samples
Number of soil samples
with Chemical A
Area A 450 80/0

Area B 550 60/0

The data in the table above were produced by ecologists who collected soil samples from two areas to
determine whether they were contaminated with Chemical A. Based on the table, what proportion of the
soil samples were contaminated with Chemical A?

A) 0.067
B) 0.069

C) 0.070
D) 0.072

13

Test negative Test positive Total

Has virus 30 370

Does not have virus 550 50 600
Total 580 420 1,000

The table above shows the results of a test that is designed to give a positive indicator when patients are
infected with a certain virus and a negative indicator when they are not infected. According to the results,
what is the probability that the test gives the incorrect indicator?

B) 80/0

D) 12%

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CHAPTER 24 PROBABILITY

14

Cured Not cured

Drug 90 25

Sugar Pill

The incomplete table above shows the results of a study in which doctors gave patients experiencing back
pain either a drug or a sugar pill. Three times as many patients were cured from the drug than from the
sugar pill. For every 2 patients cured by the sugar pill, 5 patients were not cured by the sugar pill. According
to the results, if a patient is given a sugar pill, what is the probability that the person will be cured of back

pain?

4
2
7
3
10
2
5

15

Gym equipment Computers Total

Juniors 240 300 540

Seniors 160
Total 460

The principal of a school is deciding whether to spend a budget surplus on new gym equipment or computers.
The incomplete table above summarizes the preferences among junior and senior class students. If a senior

from the school is chosen at random, the probability that the student prefers gym equipment is —. How
many seniors are at the school?

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 Statistics I
Consider this list of numbers:
562227
The mean of the list is the average:
5+6 +2+2+2+7
6

The median is the number in the middle when the list is in order. For example, the median for 1, 2, 3, 4, 5 is 3.

For our particular list, which looks like

222567
when ordered, there is no single middle number we can consider the median. When that happens, the median
is the average the two middle numbers:
2+5 3.5
2

Now what if the list were 100 numbers long? How would you determine the median? Take half to get 50. The
50th and 51st numbers would be the ones in the middle you would average.

For an ordered list of 101 numbers, take half to get 50.5. Round up. The 51st number is the median.
Seems a little counterintuitive, right? If you find this hard to memorize, just keep the smallest case in your
back pocket. For a list of 3 numbers, the second one is obviously the median. How would we get this
mathematically? Take half of 3 to get 1.5. Round up to 2, which designates the second number. For a list of 4
numbers, the median is the average of the second and third numbers. Take half of 4 to get 2. This designates
the second and third numbers.

In both cases, we "rounded up." When there was an odd number of numbers, we rounded 1.5 up to 2. When
there was an even number of numbers, we rounded 2 up to 3, which indicated that two numbers would
contribute to the median. This technique may seem a bit odd, but many students have found it helpful in
quickly finding the median of a large batch of numbers.

The mode is the number that shows up the most often. In our particular list, it's@.

The range is the difference between the biggest number in the list and the smallest number:

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CHAPTER 25 STATISTICS 1

The standard deviation is a measure of how spread out a list of numbers is. In other words, how much they
"deviate" from the mean. The standard deviation is lower when more numbers are closer to the mean. The
standard deviation is higher when more numbers are spread out away from the mean. For example, our
list

2225
would have a higher standard deviation than the following list

555567
because the second list is more around the mean. It turns out that the standard deviation
tightly clustered
of our list is 2.28 and the standard deviation of the second list is 0.83. Don't worry about how we got these
values—you'll never be asked to calculate the standard deviation on the SAT. Just know how to compare one
list's standard deviation with another's as we just did.

EXAMPLE 1:
HoursSpent Pl+Eing Sports

3
H0UV$

the -histograrmabove summarizes the daily number ofrhours spent•playing sports for 80 sttååents ata
school.

PART V: What is themean•daily number of hours spent playing sports 80 students?

ART 2ö.aWh@t the median daily number Of hours Spent playing sports for '!hé 80 StudentS?

Part 1 Solution: Sum up the total number of hours for every student. Then divide that by the number of
students.
Total hours (o x 5) + (1 x 35) + (2 x 15) + (3 x 25) 140
1.75
Number of students 80 80

and 41st students are the two in the middle (the histogram
Part 2 Solution: In a group of 80 students, the 40th
already orders the students by their hours so we
to). The first 5 students spend 0 hours playing
don't have
sports each day. The next 35 students spend 1 hour. This group includes the 40th student, so the 40th student
spends 1 hour. The next 15 students spend 2 hours. Now this group includes the 41st student, so the 41st
student spends 2 hours. Taking the average,

Daily hours spent by 40th student + Daily hours spent by 41st student
1.5
2 2

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 THE COLLEGE PANDA

EXAMPLE E

o 2
Flights Taken
3456
m a Year

The dot plot above summarizes t]vnurnber of flights taken in a year by 19 college students. If the student
who took 6 flights in a year is removed from data. which of the following correctly descrü'es the
changes to statistical nu!asures of data?

The rnean decreases.

II. median decreases.
Ill. range decreases.

A) only

B) land 11 only

C) I and
D) 111

The student who tixmk 6 flights in a year is called an outlier, an extreme data point that is far outside where
most of the data lies, Because this outlier is greater than the rest Of the data, it brings the average (mean) up. It
also increases the range since there is a larger gap between the minimum (0) and the maximum

When this outlier is removed, the mean decreases and the range decreases. The median, however, is unaffected.
To confirm this, let's calculate it, removed, there are 19 students, and the median is
Before the outlier is

represented by the 10th student, who took one flight. is removed. there are 18 students. and
After the outlier
the median is represented by the 9th and 10th students, both Of whom took One flight. so the median Of I does

not change, And in fact, outliers typically affect the mean but not the median. Answer (C)

EXAMPLE 3: Theaverage weight Of a group Of pandas is 200 pounds—Another panda/weighinB 230

pounds, joins the group,taisingthe average weightot the entire group to 205 pounds. How tnanypandas
wereiit the original group?

Once you will get a word problem that involves averages. These questions have less to do with
in a while,
and more to do with algebra, but because we cover averages in this chapter, we decided to cover
statistics
these types of word problems here as well.

When dealing with average questions on the SAT, think in terms of sums or totals. You can always find the
sum by multiplying the average with the number of subjects.

l.ß*t the number Of pandas in the original group be x, The total weight of the original group is then When
another panda joins the group, the number of pandas is x +1 and the total weight is 205(x + 1).

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CHAPTER 25 STATISTICS 1

Since that panda weighs 230 pounds,

200x + 230 205(x 1

200x + 230 205x 205

-25

There were 5 pandas in the original group.

EXAMPLE 4:
Neighborhood A

30

20

0123 4 5 6
Number Of cars owned
Neighborhood B

10

0123456
Number of cars owned
The bar charts above summarize the number of cars that residents from two neighborhoods, A and Br
own. Which of thefollowing correctly compares the standard deviation of thenumber of cars owned by
iBidents in each of the neighborhoods?

A)- The standard deviation of the number of cars owned by residents in Neighborhood A Margen
B) The standard deviationdf thenumber of cars owned by residents in Neighborhood B is larger.

C) The standard deviation of thenumber of cars owned by residents in Neighborhood A and

Neighborhood is the same;
D) The relationship cannot be determined from the information given.

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 THE COLLEGE PANDA

Most of the data for Neighborhood B are at the ends and are much more spread out from the mean, which,
because the bar graph is symmetrical, we can estimate to be 3 cars. The data for Neighborhood A, on the
other hand, are more clustered towards the low end, where the mean is. Therefore, the standard deviation for

Neighborhood B is larger. Answer (B)

CHAPTER EXERCISE: Answers for this chapter start on page 310.

A calculator is allowed on the following

questions.
6

The average height of 14 students in one class is O

63 inches. The average height of 21 students in
another class is 68. If the two classes are o
combined, what is the average height, in inches,
of the students in the combined class?

64.5 z
B) 65
C) 66
D) 665
Books read

books read last year by 20 editors at a publishing

has taken five tests in science class. The
Kristie company. Which of the following could be the
average of all five of Kristie's test scores is 94. median number of books read by the 20 editors?
The average of her last three test scores is 92.
What is the average of her first two test scores?
B) 12
A) 95
C) 17
B) 96
D) 22
C) 97
D) 98

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CHAPTER 25 STATISTICS 1

Miss World Titleholders Locks are sections of canals in which the water
level can be mechanically changed to raise and
lower boats. The table below shows the number
of locks for 10 canals in France.

Name # Locks

Aisne 27

18 19 20 21 22 23 24 Alsace 25

Age (years) NIone 5

Centre 30
The dotplot above shows the distribution of ages
Garonne 23
for 24 winners of the Miss World beauty pageant
at the time they were crowned. Based on the Lalinde 27
data, which of the following is closest to the Midi 32
average (arithmetic mean) age of the winning Oise 27
Miss World pageant contestant?
Vosges 93
A) 19 Sambre 29
B) 20
Removing which of the following two canals
C) 21
from the data would result in the greatest
D) 22 decrease in the standard deviation of the
number of locks in each canal?

A) Aisne and Lalinde

B) Alsace and Garonne
C) Centre and Midi

D) Rhone and Vosges

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The tables below give the distribution of travel 10

times between two towns for Bus A and Bus B 9 Company A
over the same 40 days. 8 Company B
Bus A 7

Travel time (minutes) 6

Frequency
o 5
44 5
4
45 10
3
47 15 z 2
48 10
1

Bus B 45 46 47 48 49

Travel time (minutes) Weight (in pounds)

Frequency
25 5
The bar chart above shows the distribution of
30 10 weights (to the nearest pound) for 19 kayaks
35 15 made by Company A and 19 kayaks made by
40 10 Company B. Which of the following correctly
compares the median weight of the kayaks made
Which of the following statements is true about by each company?
the data shown for these 40 days?
A) The median weight of the kayaks made by
A) The standard deviation of travel times for Company A is smaller.
Bus A is smaller. B) The median weight of the kayaks made by
B) The standard deviation of travel times for Company B is smaller.
Bus B is smaller. C) The median weight of the kayaks is the
C) The standard deviation of travel times is same for both companies.
the same for Bus A and Bus B.
D) The relationship cannot be determined
D) The standard deviation of travel times for from the information given.
Bus A and Bus B cannot be compared with
the data provided.

Temperature (OF) Frequency

60 3
61 4
63 4
10

70 7

The table above gives the distribution of low

temperatures for a city over 28 days. What is the
median low temperature, in degrees Fahrenheit
(OF), of the city for these 28 days?

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CHAPTER 25 STATISTICS 1

11

A shoe store surveyed a random sample of 50 5

customers to better estimate which shoe sizes
should kept in stock. The store found that the 4
median shoe size of the customers in the sample
is 10 inches. Which of the following statements 3
must be true?
A) The sum of all the shoe sizes in the sample 2
is 500 inches.
1
B) The average of the smallest shoe size and
the largest shoe size in the sample is 10
inches.
5 6 7 8 9 10
C) The difference between the smallest shoe
Integers
size and the largest shoe size in the sample
is 10 inches. The graph above shows the frequency
D) At least half of the customers in the sample distribution of a list of randomly generated
have shoe sizes greater than or equal to 10 integers between 5 and 10. Which of the
inches. following correctly gives the mean and the range
of the list of integers?

10 A) Mean 7.6, Range 4

B) Mean 7.6, Range

A food company hires an independent research C) Mean 8.2, Range
agency to determine its product's shelf life, the D) Mean 5
8.2, Range
length of time it may be stored before it expires.
Using a random sample of 40 units of the
product, the research agency finds that the
product's shelf life has a range of 3 days. Which
of the following must be true about the units in
the sample?
Quiz 1 2 34 5 6 7

Score 87 75 90 83 98 87 91
A) All the units expired within 3 days.
B) The unit with the longest shelf life took 3 The table above shows the scores for Jay's first

days longer to expire than the unit with the seven math quizzes. Which of the following are
true about his scores?
shortest shelf life.

C) The mean shelf life of the units is 3 more I. The mode is greater than the median.
than the median. II. The median is greater than the mean.
D) The median shelf life of the units is 3 more
111. The range is greater than 20.
than the mean. A) II only

B) 111 only

C) 11 and 111

D) 1, 11, and 111

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13 14

Calories in Meals
School A
5 " School B 500 500 520 550 550
550 550 600 600 900

The table above lists the number of calories in

O 3
each of Mary's last 10 meals. If a 900-calorie
meal that she had today is added to the values
listed, which of the following statistical
z measures of the data will not change?
1
I. Median
II. Mode
1 2 3 4 5 111. Range
Number of films shown
A) I and II only
The bar chart above shows the number of films B) 1 and 111 only
shown in class over the past year for 19 classes C) II and Ill only
in School A and 15 classes in School B. Which of
D) 1, 11, and 111
the following correctly compares the mean and
median number of films shown in each class for
the two schools? 15
A) The mean and median number of films
shown in each class are both greater in
School A.
B) The mean and median number of films
shown in each class are both greater in
School B.
C) The mean number of films shown in each 21 22 23 24 25 26 27 28 29 30
class is greater in School A, but the median
Gas mileage (miles per gallon)
is the same in both schools.
D) The mean and median number of films The dotplot above gives the gas mileage (in

shown in each class are the same in both miles per gallon) of 15 different cars. If the dot
schools. representing the car with the greatest gas
mileageis removed from the dotplot, what will

happen to the mean, median, and standard

deviation of the new data set?
A) Only the mean will decrease.
B) Only the mean and standard deviation will
decrease.

C) Only the mean and median will decrease.

D) The mean, median, and standard deviation
will decrease.

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CHAPTER 25 STATISTICS 1

16

Snowfall (in inches)

45 48 49 50 52 54
55 57 57 57 58 59

60 60 61 61 65 90

The table above lists the amounts of snowfall, to

the nearest inch, experienced by 18 different
cities in the past year. The outlier measurement
of 100 inches is an error. Of the mean, median,
and range of the values listed, which will change
the most if the 90-inch measurement is replaced
by the correct measurement of 20 inches?

A) Mean
B) Median
C) Range
D) None of them will change.

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 26
Statistics 11
The goal of statistics is to be able to make predictions and estimations based on limited time and information.
For example, a statistician might want to estimate the mean weight of all female raccoons in the United
States. The problem is that it's impossible to survey the entire female raccoon population. In fact, by the
time that could be accomplished, not only would the data be out of date but there would be new females in
the population. Instead, a statistician takes a random sample of female raccoons to make an estimation of
what the actual mean might be. In other words, the sample mean is used to estimate the population mean.
Using a sample to predict something about the entire population is a common theme in statistics and in SAT

questions.

food store chose customers at randot€and asked eachCUstomer I-vow manypetS

he;oEshe The results are *hown in the table below.
'Ndmbér Of PetS NUmber Of CUStomerS
600

4 or;more 100

There area tOtalof 18,000 customers in thestore'sdatabäSe. Based on the survey data, what isthe expected
total number Ofcustomers who own 2 pets?

Using the sample data, we can estimate the total number who own 2 pets to be

200
18, ooo x 1,OOO ¯ 3, 600

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CHAPTER26 STATISTICS 11

EXAMPLE 2:
Oxygen Uptake versus Heart Rate
3

23

15

100 110 120 130

Heart rate (t*'ats per minute)

The scatterplot above shows tlu• relationship between heart rate and oxygen uptake at 16 different points
during Kyle's exercise routine. line Of best fit is also shown.

PART I: Based on the line of fit, what is Kyle's predicted oxygen uptake at a heart rate of 110 beats
per minute?

PART 2: What is oxygen uptake, in liters per minute, of measurement represented by data
point that is farthest from the line of best fit?

Part 1 Solution: Using the line of best fit, we can see that at a heart rate of 110 beats per minute (along the

x-axis), the oxygen uptake is •1.5 liters per minute.

Using the line of best fit to make a prediction can be dangerous, especially when
we are making a prediction outside the scope of our data set (predicting the oxygen uptake at a heart rate
of 250 beats per minute, for example—you'd probably be

there are outliers that may heavily influence the line of best fit (see Part 2).

the data is better modeled by a quadratic or exponential curve rather than a linear one. In this case, a
linear model looks to be the right one, but something like compound interest may look linear al first even
though it's exponential growth,

Part 2 Solution: From the scatterplot, we can see that the data point farthest away from the line Of best fit is at
118 along the x-axis, The point represents an oxygen uptake of 2.5 liters per minute.

Note that this data point is likely an outlier, which can heavily influence the line Of best fit and throw Off Our
predictions. Outliers should be removed from the data if they represent special cases or exceptions,

Not only will you be asked to make predictions using the line of best fit, but you'll also be asked to interpret
its slope and y-intercept, We'll use the data from this example in the next one to show you how these concepts

are tested.

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 THE COLLEGE PANDA

EXAMPLE S:
Oxygen Uptake versus Heart Rate
3

E 25

5 0.5

90 110 120 13m

Heart rate (beats per minute)

The scatterplot shows the relationship between heartrate and 0*ygen Uptake at 16 different points
during Kyle's exercise routine, The line of bestflt'is also shown,

PARTÄ: Which Of the following is the best interpretation of the slope of Che line of best fit in the context
of this problem?

A). The predicted increase m

Kyle's oxygen uptake, m titers per mmutej for everv onebé?t per minute.
Increase in his heart rate

B) The predicted Increase m

kylets heart raten in beats per minute, for evevone •ii ey Pere minute
increase in Oxygen uptake
C) Kyle's predicted oxygen uptakeqn liters per minute at rate of 0 beats perminüte

b) Kyle's predicted heart rate in beats per ynifiute at an oxygen uptakesof Oliters per minute

PART o! t}vefoUOWing is the best interpretatiOn of they-inteteept Of th€line of best fit<in the.
Context Of Problem?

*The Predicted increase in Kyle's oxygen uptakéEinJitersper minuté„foi every one beat minute
Increase In his heart rate

B) The predicted increase kyle's heart rate, an.beaCS per minute—for every one liter per minute
increase In hs.oxygen uptake

C) Kyle's predicted oxygen uptake-in liters per minuie at a heartrate of Ö beats.per.mmute.

D) Kyle's predictedhearerate in beaisper miniite at an oxygen uptakeeof Oiitersper minute

Part 1 Solution: As we learned in the linear model questions in the interpretation chapter, the slope is the
increase in y (oxygen uptake) for each increase in x (heart rate). The only difference now is that it's a predicted
increase. The answer is (A)

Part 2 Solution: The y-intercept is the value of y (oxygen uptake) when x (the heart beat) is 0. The answer is

(C) . Note that this value would have no significance in real life since you would be dead at a heart rate of 0.

This again illustrates the danger of predicting values outside the scope of the sample data.

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CHAPTER 26 STATISTICS 11

EXAMPLE 4: Målden is a town the sCåteof Massachusetts. A real estäte,agent randomly surveyed 50
apartments for sale in Maiden and found that theaverageprice of each apartmeni was
real estate agent intends to replicate the surveyand will attempt to geta smaller margin of error: Which
of the following samples will most a smaller margin of error for the mean price of an
likely result in
apartment in Malden, Massachusetts?

A) 30@ndomly selected apartments in Malden

B) 30 randomly selected apartments in all of Massachusetts

C) 80 randomly selected apartmentsinMalden

D) 80 randomly selected apartments in all of Massachusetts

The answer is (C) . The margin of error refers to the room for error we give to an estimate. For example, we
could say the mean price of an apartment in Malden is $150,000 with a margin of error of $10,000. This implies
that the true mean price of all apartments in Malden is likely between $140,000 and $160,000. This interval is
called a confidence interval (see Example 6).
To get a smaller margin of error in Example 4, we should first only select from apartments in Malden. Selecting
apartments from all of Massachusetts not only introduces more variability to the data but also strays from the
original intent of the survey, which is to find the average price of Malden apartments. Secondly, we should use
a larger sample size. This is common The more apartments we survey, the more accurate our data and
sense.
our estimations are and the lower our margin of error is.
In fact, the margin of error for any estimate from an experiment depends on two factors:

• Sample size
• Variability in the data (often measured by standard deviation)
The larger the sample size and the less variable the data is, the lower the margin of error. We typically can't
control the standard deviation of the data (how spread out it is), but we can control the sample size. So why
don't researchers always use huge sample sizes? Because it's too costly and time-consuming to gather data
from everyone and everywhere.

ERAMPLE 5: ResearcherS conducta an expermmene to determine whether exercise Unproves stGdént

exam scores. They-randomly•seleéted 200 Studentswho exercise atåe@ét once a week and 200 stu4ents
whodo not exercise at leastoncea week After trackingthe stddents'écademiéperformances for.year,
the@esearchers found thay the students wheexercise at least once a week performedsignificantly better
on the Same examS thaiVthe students who do n0ty Based owthe design and resultSü0f Chestudy,
the fOIJowing is an appropriate conclusiOn+

A) E*erqsing at least once a weeKiS1ikeWto inwroVe scores

B) Exercising three times a Week improves examscoresnore than e*étcismg just oncou
C) Aby student Starts exercising at least-Once a weewwill improve hiS Or her exam scores
V) There is a VOSitive asSOéiatioObetvgeen exercise and

This question deals with a classic case of association (also called correlation) vs. causation. Just because
students who exercise got better exam scores doesn't mean an improvement in exam
that exercise causes
scores. It's just associated with an improvement in exam scores. Perhaps students who exercise just have more
discipline or they have more demanding parents who make them study harder. Due to the way the experirnent
was designed, we can't tell what the underlying factor is.

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 THE COLLEGE PANDA

Therefore, answer (A) is wrong because it implies causation. Answer (B) is wrong because it not only implies
causation but also implies that the frequency of exercise matters, something that wasn't tracked in the experiment.
Answer (C) is wrong because it suggests a completely certain outcome. Even if exercise DID improve exam
scores, not every single student who starts exercising will improve their scores. There might be students for
whom exercising makes their scores worse. Any conclusion drawn from sample data is a generalization and
should not be regarded as a truth for every individual.

The answer is (D) . There is a positive association between exercise and student exam scores.

One of the things the researchers did correctly was to take random samples from each group. The key word is
random. If we wouldn't even have been able to conclude that there is a positive
the samples weren't random,
association between exercise and exam scores. Why? Let's say the researchers picked 30 students from the
tennis team for the exercise group and 30 students who just play video games all day for the non-exercise
group. Definitely not random. Now, did the exercise group do better on their exams because they exercise
or because they play tennis? Or was it the video games that made the non-exercise group perform worse?
Because the selection isn't random, we can't tell how each factor influences the result. When the selection is

random, all the factors except the one we'æ testing are "averaged out."
Now what if the researchers wanted to see whether exercise does indeed cause an improvement in exam scores.
What should they have done differently? The answer is random assignment. Instead of randomly selecting 200
students from one group that already exercises regularly and 200 students from another group that does not,
they should have just randomly selected 400 students. The next step would be to randomly assign each student
to exercise or not. Everyone in the exercise group is forced to exercise at least once a week and everyone in the
non-exercise group is not allowed to exercise. If the exercise group performs better on the exams, then we can
conclude that exercise causes an improvement in exam scores. Of course, conducting this type of experiment
can be extremely difficult, which is why proving causation can be such a monumental task.

The following list summarizes the conclusions you can draw from different experimental designs involving
two variables (e.g. exercise and exam scores).
1. Subjects not selected at random & Subjects not randomly assigned
• Results cannot be generalized to the population.

• Cause and effect cannot be proven.

e Example: Researchers want to see whether medication X is effective in treating the flu. People
with the flu from Town A receive medication X. People with the flu from Town B receive a placebo
(sugar pill). More people in the medication X group experience a reduction in flu symptoms. The
generalizafion that medication X is associated with a reduction in flu symptoms cannot be made
since was only tested in Town A and Town B (sample was not randomly selected from the general
it

population). There may be something special about Town A and Town B. No cause and effect
relationship can be established because the medication was not randomly assigned. Perhaps Town
A experienced a less severe flu epidemic.
2. Subjects not selected at random & Subjects randomly assigned
• Results cannot be generalized to the population.

• Cause and effect can be proven.

• Example: Researchers want to see whether medication X is effective in treating the flu. People
with the flu A and Town B are randomly assigned to either medication X or a placebo
from Town
(sugar pill). More people in the medication X group experience a reduction in flu symptoms. The
generalization that medication X is effective for everyone cannot be made since it was only tested
in Town A and Town B (sample was not randomly selected from the general population). Perhaps
only one particular strain of the flu exists in Town A and Town B. A cause and effect relationship
can be established because the medication was randomly assigned. For the people in Town A and

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CHAPTER 26 STATISTICS 11

Town B, we can conclude that medication X causes a reduction in flu symptoms. Note that this is

still just a generalization—as with any other medication, medication X does not guarantee you will
definitely get better, even if you live in Town A or Town B.
3. Subjects selected at random & Subjects not randomly assigned
• Results can be generalized to the population.

• Cause and effect cannot be proven.

• Example: Researchers want to see whether medication X is effective in treating the flu. People
with the flu from the general population are randomly selected. They are given the choice of a
new medication (medication X) or a traditional medication (really a sugar pill). More people in the
medication X group experience a reduction in flu symptoms. We can generalize that people who
choose to receive medication X fare better than those who don't. However, no cause and effect
relationship can be established because the medication was not randomly assigned. We don't know
whether the reduction in symptoms is due to the medication or a difference between those who
volunteered and those who didn't.

4. Subjects selected at random & Subjects randomly assigned

• Results can be generalized to the population.

• Cause and effect can be proven.

• Example: Researchers want to see whether medication X is effective in treating the flu. People
with the from the general population are randomly selected. Using a coin toss (heads or tails),
flu
researchers randomly assign each person to either medication X or a placebo (sugar pill). More
people in the medication X group experience a reduction in flu symptoms. We can conclude that
medication X causes a reduction in flu symptoms. This conclusion can be generalized to the entire
population of people with the flu.

EXAMPLE 62 Environmentalists are testing pHlevelSiOæforestthatis bein harmedbraacid rain. They*

analyze4Vater samples from 40 rainfalls inghe past yearand found that the mean pH theqipter$ample$
has a 95% eojlfidence interval Of 3.2@3$. Whicmof the follOwin COnclüs10ns iSthe 0100 appropriate
'based on„the confidence interval?"'

) 95% ofållthe forest rainfallS in thé past year have pHbetween3.2and 3.8,
B) 95%of all the fméserainfalls m the a pHbetween > and 38
C) It is praUsibJe that the Of all the-forestidififalls in yearisbetweerv3.2and

D) ItjS Plaåzsiblethatthe the past decadéjs between 3.2 and

3.8.

Ifyou don't know what a confidence interval is, don't worry. You'll never need to calculate one and the SAT
makes these questions very easy. All a confidence interval does is tell you where the true mean (or some other
statistical measure) for the population is likely to be (e.g. between 3.2 and 3.8). Even though the SAT only
brings up 95% confidence intervals, there are 97% and 99% (any percentage) confidence intervals. The higher
the confidence, the more likely the true mean falls within the interval. So in the example above, we can be
quite confident that the hue mean pH of all the forest rainfalls in the past year is between 3.2 and 3.8. Answer

(C) . The answer is not (D) because we cannot draw conclusions about the past decade when all the samples
were gathered from the past year.
A confidence interval does NOT say anything about the rainfalls themselves. You cannot say that any one
rainfall has a 95% chance pH between 3.2 and 3.8, and you cannot say that 95% of all the forest
of having a
rainfalls in the past year had a pH between 3.2 and 3.8. Always remember that a confidence interval applies

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 THE COLLEGE PANDA

only to the mean, which is a statistical measurement, NOT an individual data point or a group of data
points.

Secondly, a 95% is a 95% chance it contains the true mean. Even

confidence interval does not imply that there
though confidence computed for the mean, you cannot say that the interval of 3.2 to 3.8 has a 95%
intervals are
chance of containing the true mean PH.

So what does it mean in statistics to be 95% confident in something? If the experiment were repeated again and
again, each with 40 water samples, 95% of those experiments would give us a confidence interval that contains
the true mean. In other words, the confidence interval given in the example is the result of just one experiment.
Another run of the same experiment (another 40 samples) would produce a different confidence interval. Keep
on getting these confidence intervals and 95% of them will contain the true mean. So the 95% pertains to all
the confidence intervals generated by repeated experiments, NOT the chance that any one confidence interval
contains the true mean. Again, don't worry about how confidence intervals are calculated, but be aware that
this is how "confidence" is defined in statistics.

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CHAPTER 26 STATISTICS 11

CHAPTER EXERCISE: Answers for this chapter start on page 312.

A calculator is allowed on the following

questions.

Traffic Light Violations in Various Towns

Male Shoe Size versus Age 100

13 90

12 80

11
70

60
10
50

40

30

20
30 40 50 60 70 80 90 100

10 11 12 13 14 15 16 17 18 19 20 Number of traffic lights

Age (years)

The scatterplot above shows the number of

The scatterplot above shows the relationship
traffic lights in 15 towns and the average weekly
between age, in years, and shoe size for 24 males
number of traffic light violations that occur in
between 10 and 20 years old. The line of best fit
each town. The line of best fit is also shown.
is also shown. Based on the data, how many 19
Based on the line of best fit, which of the
year old males had a shoe size greater than the
following is the predicted average weekly
one predicted by the line of best fit?
number of traffic light violations in a town with
75 traffic lights?

A) 40
B) 50
C) 55
D) 60

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A university wants to determine the dietary Consumer Behavior during Store Sales
preferences of the students in its freshman class.
60
Which of the following survey methods is most 55
likely to provide the most valid results? 50
A) Selecting a random sample of 600 students 45
40
from the university
35
B) Selecting arandom sample of 300 students 30
from the university's freshman class 25
C) Selecting a random sample of 600 students 20
from the university's freshman class 15
10
D) Selecting a random sample of 600 students Q.) 5
from one of the university's freshman
dining halls 0 5 10 15 20 25 30 35 40 45 50
Store Discount (%)

Shopping time refers to the time a customer

spends in one store. The scatterplot above shows
Two candidates are running for governor of a the average shopping time, in minutes, of
A recent poll reports that out of a random
state. customers at 26 different stores offering various
sample of 250 voters, 110 support Candidate A discounts. The line of best fit is also shown.
and 140 support Candidate B. An estimated Which of the following is the best interpretation
500,000 state residents are expected to vote on of the meaning of the y-intercept of the line of
election day. According to the poll, Candidate B best fit?

is expected to receive how many more votes

A) The predicted average shopping time, in
than Candidate A?
minutes, of customers at a store offering no
A) 60,000 discount

B) 130,000 B) The predicted average shopping time, in

C) 220,000 minutes, of customers at a store offering a
50% discount
D) 280,000
C) The predicted increase in the average
shopping time, in minutes, for each one
percent increase in the store discount

D) The predicted average number of

customers at a store offering no discount

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CHAPTER 26 STATISTICS 11

Advertising for 16 Companies Movie Length versus Box Office Sales

500 200
450 180
o
400 o 160
o
350 140
300 1.
120
250 100
O
200 80
150 60
100 40
50 20
0 0
0 10 20 30 40 50 60 70 80 90 100 o 60 70 80 90 100 110 120 130 140 150
Advertising Expenses (in thousands of dollars) Movie Length (minutes)

The scatterplot above shows the relationship The scatterplot above plots the lengths of 15
between revenue and advertising expenses for movies against their box office sales. The line of
16 companies. The line of best fit is also shown. best fit is also shown. Which of the following is
Which of the following is the best interpretation the best interpretation of the meaning of the
of the meaning of the slope of the line of best fit? slope of the line of best fit?

A) The expected increase in revenue for every A) The expected decrease in box office sales
one dollar increase in advertising expenses per minute increase in movie length
B) The expected increase in revenue for every B) The expected increase in box office sales per
one thousand dollar increase in advertising minute increase in movie length
expenses C) The expected decrease in box office sales
C) The expected increase in advertising per 10-minute increase in movie length
expenses for every one thousand dollar
D) The expected increase in box office sales per
increase in revenue
10-minute increase in movie length
D) The expected revenue of a company that
has no advertising expenses

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Mistakes Made in Incentive-based Task Fat and Calories of Ice Cream

60 440

50 420

400
40
380
30
360
o
20

10 320
z
o 100 200 300 400 500 600 15 20 25 30 35
Prize (in dollars) Total fat (grams)

In a psychological study, researchers asked The above shows the fat content and
scatterplot
participants to each complete a difficult task for calorie counts of 8 different cups of ice cream.
a cash prize, the amount of which varied from Based on the line of best fit to the data shown,
participant to participant. The results of the what is the expected increase in thenumber of
study, as well as the line of best fit, are shown in calories for each additional gram of fat in a cup
the scatterplot above. Which of the following is of ice cream?
the best interpretation of the meaning of the
y-intercept of the line of best fit?

A) The expected decrease in the number of

C) 20
mistakes made per dollar increase in the
cash prize D) 40

B) The expected increase in the number of

mistakes made per dollar increase in the
cash prize
C) The expected dollar amount of the cash
prize required for a person to complete the
task with 0 mistakes

D) The expected number of mistakes a person

makes in completing the task when no cash
prize is offered

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CHAPTER 26 STATISTICS 11

10 11

Nitrogen Fertilizer and Oats Food Courts in Various Malls

80

70
80
60

50 60

40
40
30
o
o 20
20
10

o 100 200 300 400 500 4 5 6 7 8 9 10

Amount of nitrogen applied (pounds per acre) Number of restaurants

The scatterplot above shows the distribution of

The scatterplot above shows the amount of
seats for the restaurants in 7 different mall food
nitrogen fertilizer applied to 8 oat fields and
The line of best fit is also shown.
courts.
their yields. The line of best fit is also shown.
According to the data, what is the total number
Which of the following is closest to the amount
of seats at the food court represented by the data
of nitrogen applied, in pounds per acre, to the
point that is farthest from the line of best fit?
oat field whose yield is best predicted by the line
of best fit? A) 200
A) 200 B) 240

B) 350 C) 320

C) 400 D) 560
D) 450
12

Researchers must conduct an experiment to see

whether a new vaccine is effective in relieving
certain allergies. They have selected a random
sample of 100 allergy patients. Some of the
patients are assigned to the new vaccine while
the rest are assigned to the traditional treatment.
Which of the following methods of assigning
each patient's treatment is most likely to lead to
a reliable conclusion about the effectiveness of
the new vaccine?
A) Females are assigned to the new vaccine.
B) Those who have more than one allergy are
assigned to the new vaccine.
C) The patients divide themselves evenly into
two groups. A coin is tossed to decide
which group receives the new vaccine.
D) Each patient is assigned a random number.
Those with an even number are assigned to
the new vaccine.

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A basketball manufacturer selects a random The length of a blue-spotted salamander's tail

sample of its basketballs each week to ensure a can be used to estimate its age. A biologist
consistent air pressure within them is selects 80 blue-spotted salamanders at random
maintained. In Week 1, the sample had a mea-II and finds that the average length of their tails

air pressure of 8.2 psi (pounds per square inch) has a 95% confidence interval of 5 to 6 inches.
and a margin of error of 0.1 psi. In Week 2, the Which of the following conclusions is the most
sample had a mean air pressure of 7.7 psi and a appropriate based on the confidence interval?
margin of error of 0.3 psi. Based on these results,
A) 95% of all blue-spotted salamanders have a
which of the following is a reasonable
tail that is between 5 and 6 inches in length
conclusion?
B) 95% of all salamanders have a tail that is
A) Most of the basketballs produced in Week 1 between 5 and 6 inches in length
had an air pressure under 8.2 psi, whereas
C) The true average length of the tails of all
most of the basketballs produced in Week 2
blue-spotted salamanders is likely between
had an air pressuæ under 7.7 psi.
5 and 6 inches.
B) The mean air pressure of all the basketballs
D) The true average length of the tails of all
produced in Week 1 was 0.5 psi more than
salamanders is likely between 5 and 6
the mean air pressure of all the basketballs
inches.
produced in Week 2.
C) The number of basketballs in the Week 1
sample was more than the number of 16
basketballs in the Week 2 sample.
D) It is very likely that the mean air pressure An economist conducted research to determine
of all the basketballs produced in Week 1 whether there is a relationship between the price
was less than the mean air pressure of all of food and population density. He collected
the basketballs produced in Week 2. data from a random sample of 100 U.S. cities and
found significant evidence that the price of food
is lower in places with a high population
14
best supported by these results?
A student is assigned to conduct a survey to A) In U.S. cities, there is a positive association
determine the mean number of servings of between the price of food and population
vegetables eaten by a certain group of people density.
each day. The student has not yet decided which
B) In U.S. cities, there is a negative association
group of people will be the focus of this survey.
between the price of food and population
Selecting a random sample from which of the
density.
following groups would most likely give the
smallest margin of error? C) In U.S. cities, a decrease in the price of food
is caused by an increase in the population
A) Residents of the same city
density.
B) Customers of a certain restaurant
D) In U.S. cities, an increase in the population
C) Viewers of the same television show density is caused by a decrease in the price
of food.
D) Students who are following the same daily
diet plan

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 27 Volume
The volume of all regular solids can be found using the following formula:

Voluines Aiea ofhaSO< héight

That's why the volume of a cube is V — s3 (the area of the base is s2 and the height is s)

The volume of a rectangular box/ prism is V lwh (the area of the base is lw and the height is h)

And the volume of a cylinder is V — Ttr2h (the area of the base is nr2 and the height is h)

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 THE COLLEGE PANDA

Even though the SAT gives you these formulas at the beginning of each math section, they should be memorized,
in addition to the volume of a cone

and the volume of a sphere

3

But what if we have a hollowed-out cylinder? What's the volume of that?

Well, if we look at the base, it's just a ring.

The area of the ring is the outer circle minus the inner circle.

7TR2 — 7tr2 — 7T(R2 — r

2)

To get the volume, we multiply this area by the height.

V— 7T(R2 — r2)h

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CHAPTER 27 VOLUME

CHAPTER EXERCISE: Answers for this chapter start on page 314.

A calculator is allowed on the following

questions.
What is the volume of a cube with surface area

B) 8a2

C) 8a3
D) 16a3
5 cm

A cylindrical water tank with a base radius of 4

feet and a height of 6 feet can be filled in 3 hours.
4 cm
At that rate, how many hours will it take to fill a
In the figure above, a cylindrical block of wood cylindrical water tank with a base radius of 6

is sliced two pieces as shown by the dashed

into feet and a height of 8 feet?

curve. What is the volume of the top piece in

A) 4.5
cubic centimeters?

A) 107T
C) 7.5
B) 157t

C) 207T
D) 407t

James wants to cover a rectangular box with

wrapping paper. The box has a square base with
an area of 25 square inches. The volume of the
box is 100 cubic inches. How many square
inches of wrapping paper will James need to
exactly cover all faces of the box, including the
top and the bottom?

A) 120
B) 130
C) 150
D) 160

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 THE COLLEGE PANDA

A cube with a side length of 5 inches is painted

black on all six faces. The entire cube is then cut
2 into smaller cubes with sides of 1 inch. How
many small cubes do not have any black paint
on them?
A) 27
B) 31

C) 36
D) 48

Yuna finds a box with an open top. Each side is 8

inches long. If she fills this box with identical 2
A container in the shape of a right circular in by 2 in by 2 in cubes, how many of these
cylinder shown above is just large enough to fit
cubes will be touching the box?
exactly 3 tennis balls each with a radius of 2
inches. If the container were emptied out and A) 40
filled to the top with water, what would be the B) 48
volume of water, in cubic inches, held by the C) 52
container?
D) 56
A) 167t

B) 247T

C) 327t
D) 487t A 3 x 4 x 5 solid block is made up of 1 x 1 x 1
unit cubes. The outside surface of the block is
painted black. How many unit cubes have
exactly one face painted black?

An aquarium has an 80 inch by 25 inch A) 16

rectangular base and a height of 30 inches. The B) 18
aquarium is filled with water to a depth of 20 C) 20
inches. If a solid block with a volume of 5,000 in 3
D) 22
completely submerged in the aquarium, by
is

how many inches does the water level rise?

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CHAPTER 27 VOLUME

10

9 cm
3

8
10
15 cm
A crate that is 10 inches long, 8 inches wide, and
3 inches high is shown
above. The floor and the
four walls are one inch thick. How many
all

A food manufacturer produces packages of one-inch cubical blocks can fit inside the crate?
frozen ice cream cones. Each ice cream cone
A) 84
consists of a right circular cone that is filled with
B) 96
icecream until a hemisphere is formed above the
cone as shown in the figure above. The right C) 120
circular cone has a base radius of 9 cm and a D) 144
slant height of 15 cm. What is the volume of ice
cream, in cubic centimeters, the manufacturer
uses for each ice cream cone? 12

A) 7297T
B) 8107T

C) 8917T
D) 9607T

5 6

Note: Figure not drawn to scale.

The concrete staircase shown above is built from

a rectangular base that is 5 meters long and 6
meters wide. The three steps have equal
dimensions and each one has a rise of 0.2 meters.
If the density of concrete is 130 kilograms per
cubic meter, what is the mass of the concrete
staircase in kilograms? (Density is mass divided

by volume)
A) 1,420
B) 1,560

C) 1,820
D) 2,040

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 Answers to the
28
Exercises
Chapter l: Exponents & Radicals
EXERCISE 1:

11. —36 20. 9

2. —1 12. 64 1
21.
9
13. -72
22. 125
14. 108
1
23.
15. -648 125

16. 1 24. 49

1
7. -1 1
25.
6 49

1 26. 1000
18.
9. ¯ 27 4
1
27.
10. 27 19. 1 1, ooo

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CHAPTER 28 ANSWERS TO THE EXERCISES

EXERCISE 2:

1. 6x5 10. x3 19. 36m 8

8 x6 1
2. 11. 20.
a6

3. 15x 2 2 21. b12

311
12.
4
4. -21 m4
22.
n
13. —8u3v3
1
5.
8X6
14. x5 23. x2

9b5 1
6. 15. 3x8 24.
mn2
16. x
n4 25. k
7.
2 17.
m6
26.
8. a4b6 2 n9
18

27. x y z
9.
x

EXERCISE 3:

13.

14.

9. 2V2 15. x — 21
1
10. 16. x
2
11. x 50 17.

12. 18.

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 THE COLLEGE PANDA

CHAPTER EXERCISE:

6. Multiply both equations together. The left

hand side gives x5y5. The right hand side

1
gives 80.

1 7. @To avoid any trickiness, it's best to plug in

1
numbers. Let a 2 and b 2. Going through
each choice,

16
1

3 B) (-4)4 256

1 C) (2-2)2 —
— 16
9
D) 2-24 —2. 16 32

(B) is the largest.

2. @It's obvious that there will be a bunch of
I's, but how many? Well, how many even 8. @The 2a means raised to the 2a power and
numbers ate there between 2 and 50? If we the b on the bottom means the bth root.
take the list
9. @Cube both sides of the first equation,
, 48, 50
( 23 = (y3)3

and divide each element by 2, 9

123 , 24,25
Now y9 can be replaced by x6,
we can clearly see that there are 25 numbers.
Therefore, n is the sum of twenty-five I's. The x 3z
answer is 25.
x
3z
x6
3z =6

22(2n+3) 23(n+5)

2(2n +3) = 301 +5)

10.
= + 15
2X+3 2X = k(2X)
(21) (23) - = k(2X)
4.

2X(23— 1) = k(2X)
2X
23 ? (7) k(2X)

2X-Y 23

3
11.

1 3 1 3

10 10 3
Therefore, a
33 33 4

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CHAPTER 28 ANSWERS TO THE EXERCISES

12. @Squaring both sides ("unsimplifying" will

get you the same result),

(2 x+2)2
— 18
— 18
4x — 10
x = 25

13. c

ac bc 30

ac+bc 30
x
ac + bc 30

(a + b)c 30
5c 30
6

14. 8, 000 Multiply the first equation by n to get

4
n nx

Substitute this into the left side of the second

equation,
nx 20x

n 20

Using the first equation,

— (20)3 8, ooo

15.

Multiply both sides by xy,

3xy

We do this to make the following substitution,

3xy 333

xy — 111

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 THE COLLEGE PANDA

Chapter 2: Percent
CHAPTER EXERCISE:

1. @Letz — 100. Then x 1.50(100) — 150 9. number of pistachios at the

Let x be the
andy — 1.20(100) — 120. x is start. At the end of each day, what's left is
1 — 0.40 = 0.60 of the day's starting amount.
150 - 120 30 Over two days,
120 120
— 27
larger than y. 0.36x

2. Each year, Veronica keeps whatever she

x — 75
has in her account plus the interest on that
amount. Because m is a percentage, we can
10. @Let x be the sales tax (as a decimal for
convert it to a decimal by dividing it by 100,
now). We'll convert it to a percent at the end.
giving us 0.01m. Therefore, x 1 + 0.01m.
— 100
3. 2 Note that the sample size of 400 is
irrelevant information. To make things easier, 100
we'll let x be a decimal for now and convert it (105.82) (.90)
to a percentage later, 100
-1 0.05 —
3,300x 66 (105.82) (.90)

x 0.02 —

53
4. C Week 3 accounted for 0.32 - 320/0
167
70
of the total box spring units sold.
56 B
103
0.267 26.7%
386
12. C Kyle ate 20(1.20) 24 pounds of chicken
6• 0.553 55.3% wings and 15(1.40) pounds of hot dogs.
21
That's a total of 24 + 21 45 pounds of food.
43 - 68 John had 20 + 15
0.37 — 37% decrease. 35 pounds of food. The
68 percent increase from John to Kyle is

8. @Let the original price of the book be $100. 45 — 35

Then James bought the book at .29 29%
100(1 - - 0.30) — 35

56
$56, which is 56% of the original price.
100 13. C Let her starting card count be x. A loss of
18 percent reduces her total to (0.82) x. From
there, an increase of 36 percent gets the total

to (1.36) (0.82)x. Now,

(1.36) (0.82)x =n
n
x
(1.36) (0.82)

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CHAPTER 28 ANSWERS TO THE EXERCISES

14. 6,460.

15. @Simple interest means that Kyle will

receive the same amount each year based on
his initial investment of $2,000. He'll receive
2, 000(0.06) in interest each year for a total of
2, 000(0.06)t after t years. He'll also get back
his initial investment. Therefore, the total
amount he receives after t years is

2, ooo + 2, = 2, ooo(l + 0.060

16. 100 Since scarves and ties make up 80% of

the accessories, the 40 belts must account for
20%. Letting the total number of accessories
be x,
200/0 of x 40

1
-x 40
5

x = 200
There are 200 accessories in the store.

Hopefully you're able to get this without

having to make an equation, but there's no
harm in a little algebra! Now we can
determine that there are >< 200 — 40 scarves

and — x 200 120 ties. Half of the 120 ties

(60 ties) are replaced with scarves, so the store

will end up with 40 + 60 — 100 scarves.
17. @The total amount in the savings account
after 5 years will be 3, but the
interest earned will be 3, — 3, 000.
The total amount in the checking account
after 5 years will be 1, but the
interest earned will be 1, — 1, 000.

With a larger deposit and a higher

initial
interest rate, obvious the savings account
it's

will have earned more interest. The difference

in earned interest will be (3,
3,000) — — 1,000).

18. @Compounded once every 2 years, interest

is earned — times in t years. The annual

interest rate of 4% must be doubled to get the

rate earned over a 2 year span. Therefore, the

correct expression is k(1.08) .

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 THE COLLEGE PANDA

Chapter 3: Exponential vs. Linear Growth

CHAPTER EXERCISE:

25 - 20
1. @With exponential growth, we need to calculate the percent increase, which turns out to be
20
0.25. Therefore, the rate is 1.25, and the growth can be modeled by P 20(1.25)t.

2. @The constant increase is 125 100 25. Therefore, the slope is 25 and the y-intercept (the initial

population) is 100.

3. @The population of trees is experiencing exponential decay at a rate of 1 — 0.04 0.96. The decrease

happens once every 4 years, so the rate should be applied times.

4. C Scatterplot C is the closest to forming a straight line.

5. C Keep track of the total amount she has received: 3, 9, 27, 81. Because the total amount she has received
triples each day, the relationship is exponential growth.

6. B Each month, Albert loses a book. Because this is a constant decrease, the relationship is linear decay
(decreasing linear).

7. The cell count doubles every hour so the rate, r, is 2. The initial count is 80 so c = 80.
8. Five percent of the original square footage is a constant. It doesn't change, which would make it

linear growth.

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 4: Proportion
CHAPTER EXERCISE:

Pold =

(o.5V)2 0.25V2
new — 0.25P01d

The electric power drops to a fourth of what it was.

2. @The area of a square is A s2, where s is the length of each side.

A new (1.10s)2 (1.10)2s2 1.21' 1.21 Aoid

The new area is 21% greater.

1
Void
3

1
new — —7T(0.80r) 2(1.10h) 0.704V01d
3 (13

The volume of the cone decreases by 1 - 0.704 — 0.296 29.6%.

1
AOId — (bl
2
+ b2)h
1
Anew C) (2) [I(bl + b2)hl — -(bl
2
+ b2)h AOId

Notice how — was factored out from bl and b2. The area stays the same.

5. B Let r be the radius of Kevin's sphere, and let x be the factor the radius of Calvin's sphere is greater by.

4
VKevin — Ttr3
3

4
VCalvin ¯ X3 — 7tr3 X VKevin
3

x3

x 1.59

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 THE COLLEGE PANDA

1 1
6. B The area of the original triangle is — (s) (s) —s 2
2 2

1
Anew —(xs)2 X Aold
2 (12

x — .80

s must have been decreased by 1 .80 0.20 20%.

Lother star —
¯ 47Td2b
Lstar 47T(3d)2(2b) (47Td2b) 18L0ther star

8. Let x be the fraction that Star A's distance is of Star B's.

1
LstarA — —LstarB
9
1
—(47Td2b)
9

1
x2 (47Td2b) — —(47Td2b)
9

2 1
x
9
1

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 5: Rates
CHAPTER EXERCISE:

xy + xy + xy — 2xy

2. For one week, Tim's diet plan would require a protein intake of 7 x 60 420 grams. Since each
protein bar provides 30 grams of protein, he would need to buy 420 + 30 — 14 protein bars.

3. [J Over 6 years, the screen size increased by a total of 18.5 15.5 3 inches. That's 3 ±6 — 0.5 inches

each year.

4. B The pressure increases by 70 50 — 20 atm while the submarine descends —900 (—700) — -200
meters. That's 20 200 = 0.1 atm per meter, or 1 atm per 10 meters.
5. 3 The pool has a capacity of 5 >< 300 1, 500 gallons. At an increased rate of 500 gallons per hour, it

would only take 1, 500 500 3 hours : to fill the pool.

d dollars 20d
20 AppleS x dollars
a AppleS a

n
7. @With ab students and n stickers, each student receives stickers.
ab

8. @The racecar burned 22 18 4 gallons of fuel in 7 —4 3 laps. To get to 6 gallons left, the racecar
will have to consume 18 —6 12 more gallons. That's

3 laps
12 gallorTS x 9 more laps
4 gallon-S

which is Lap 7 + 9 — 16.

9. took 2.5 hours for 65 —40 25 boxes to be unloaded. There are 3.5 hours from 3:30PM to 7:00PM.
25 boxes
In 3.5 hours, 3.5 hotlfS x 35 more boxes will be unloaded. That's a total of 65 + 35 100
2.5
boxes.

d 400
10. Amy spends — dollars each week on fruit. Therefore, it will take her 100 weeks to spend
4 d
$100 on fruit.

11. The setup fees amount to 100c, $100 for each customer. The monthly cost for all the customers
amounts to 50c, $50 for each customer. Over m months, the monthly charges add up to 50c x m, or
50cm. The total charge is therefore 100c + 50cm.

12. The compound's temperature increases by — degrees per minute. So after x minutes, the temperature
dx dx
increases by —x, or . The final temperature is then t +
m

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 THE COLLEGE PANDA

13. @The reduced price of each souvenir after the first is 0.6a. After the first souvenir, there are n I

souvenirs that James will purchase at the reduced price. Therefore, the total cost is a + (n 1) (0.6a).

14. D The bakers make 3xy cupcakes each day. Over 4 days, they will make a total of 4 x 3xy — 12xy
cupcakes. The number of boxes needed is the total number of cupcakes divided by the number of
12xy
cupcakes that can fit in each box: = 12y.

15. @For mn students, the total number of slices must be 2mn. Since there are 8 slices in each pizza, the

school must order

2mn
8
—
mn
4
pizzas.

16. The percent change is the new minus the old over the old times 100. Notice that the P's cancel out.

100
x 100 —1 x 100
{öö)5

17. A The first 150 miles took 150 + 30 — 5 hours. The next 200 miles took 200 50 = 4 hours. His average
:

speed, total distance over total time, was (150 + 200) / (5 + 4) 38.89 miles per hour.

18. 120 Average speed is just total distance over total time. The total distance, in inches, was 2400 x 12
28, 800. The total time, in seconds, was 4 x 60 = 240. 28, 800 + 240 = 120 inches per second.

19. 432
90 words
12 x 432 words
2.5 minutéS

20. 8 The painter can cover an area of 97T in 2 minutes. A circular region with a radius of 6 feet has
an area of 367T.

2 minutes
367T X 8 minutes
97T

21. @The clock falls behind by 8 minutes every hour. There are 6.5 hours between 4:00 AM and 10:30 AM,
so the clock falls behind by 8 x 6.5 52 minutes. The correct time is then 52 minutes past 10:30 AM,
which is 11:22 AM.

100 jg.pxoductS 1 jar

180 slon x 60 jars
15jn_com-miSSTÖK x 20 jLpreductS

23.

60minuteS 32 kilometers
2 houfS x 265 kilometers
1 hotlf

24. Two liters is equivalent to 2 x 33.8 — 67.6 ounces, which will fill 67.6 + 12 5.63 plastic cups. So at
most, 5 plastic cups can be completely filled.

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CHAPTER 28 ANSWERS TO THE EXERCISES

25. @What makes this question a little tricky is that we don't know the distance Brett travels each month or
thenumber of gallons he uses each month. Let's say he needs 2 gallons of gas each month (you can make
up any number you want). That means he travels 30 x 2 — 60 miles each month and each gallon costs
160 2 80 dollars (ridiculous, I know). Now if he switches to the new car, he'll only need 60 + 40 — 1.5

gallons of gas each month (distance of 60 miles divided by the 40 miles per gallon). Because the price of
gas stays the same, that will cost him 1.5 x 80 120 dollars each month.

26. 8 inch by 10 inch piece of cardboard has an area of 8 x 10 80 square inches. A 16 inch by 20 inch
piece of cardboard has an area of 16 x 20 = 320 square inches.

2 dollars
320 yquare-incfiéS x 8 dollars
80 ±quare-irtcfiéS

27. 48 Each jar of honey costs 9 +4 = 2.25 dollars. She can sell each jar for 15-33 = 5 dollars. That's a profit
of 5 — 2.25 2.75 dollars per jar. To make a profit of 132 dollars, she would have to sell 132 + 2.75 48
jars.

28. B Working at the slowest pace, Jason would take 100 +6 16.67 hours. Working at the fastest pace, he
would take 100 + 8 — 12.5 hours. The only answer choice between those two numbers is 16.

29.

16 ounees 29.6 my 1 student

6 gups x 28.4 students
1 y.W 1 DA*tcé 100 mL
Since it wouldn't make sense to have four-tenths of a student, the most that can be accommodated is 28
students.

30. @Jessica runs at a rate of 4 yards per second. Let t be the time it takes for Jessica to overtake Yoona. We
can make an equation with the left side being Yoona's distance and the right side being Jessica's distance.

30 +t 4t

30 3t

10 t

It takes 10 seconds for Jessica to catch up to Yoona. In that time, Jessica runs 4(10) 40 yards.

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Chapter 6: Expressions
CHAPTER EXERCISE:

1. We factor out 6xy from both terms to get 6xy(x + y).

4 3a 4 -k 3a
2. B The common denominator is 4a.

3. @Expanding,
(x2 + y) (y + z) x2y + x2z + Y2 + yz

4. @Divide the top and bottom by 4 to get

3x
Another way to get the same answer is to split the

fractions and reduce.

5. = 3(x2 -1) 3(x2 1)

6. @The expression follows the (a + a2 + 2ab + b2 pattern, where a x + 1 and b Y + 1. Therefore,

the expression is equivalent to ( (x + I) + (y + 1))2
7. XY — x2 x(y — x) —x(x — y) x
xy — Y2 y(x — Y) y(x — y)

8. Adding the two fractions in the denominator,

5x +7
2 3 6 6

6
Now, 1 over this result means we can flip it:

x x x
x
1 x 2x — 1
2
x x

10. @Combining like terms, we get 3x3 + (8x2 + 7x2) + (—4x llx) —7— 3x3 + 15x2 - 15x

11. Combining like terms, 5a — 2a -2v/å.

12.
3 36y2 + 72y2 1 3
2 72y2 2 2

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Chapter 7: Manipulating & Solving Equations

CHAPTER EXERCISE:

19.

27T

XY + XZ - X 1

X(Y+Z-I)—

—1

20.
6.
7th

7.

8. 2x ¯ Y — xy
S — 27Tr2 2x — y(l x
27tr

bc
10. a
I—x

bc
11. d 21. First, cross-multiply.

2ac ab +b
12. m 2ac ab

Y2 m(X2 — Xi) + Yl mx2 — mxl + Yl a(2c — b) —

ITtX2 — Y2 + Yl
14.
2c — b

15. a

bx
3t

47T2L
17, g
23. Divide both sides by 3 to get x + 2y
18. p — —q
7T2r2 24. Multiply both sides by 2 to get 2x + 10—@

25. Since 2t — , we can multiply both sides

by 2 to get 4t —

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26. Cross multiply.

32. a
b2 +2

3p — 311 2p 211
33.

p 5h
7X2 +3
k(x2 + 4) + ky
-B 2

7x2 +3
k(x2 y)
2
27. Cross multiply.
7x2 +3
1
2(x2 4 Y)
1

34.

28. Square both sides to get (xY)2 b

4x+1
29. p b
(x3 — x2)(x5 — x4)
b
1
x3
l)
30. m b
x —3

1
31. n

5x2 —3

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CHAPTER 28 ANSWERS TO THE EXERCISES

CHAPTER EXERCISE:

7. Cross multiply to get 12 kx + 2x. Then,

-8 12 2x
(-2)3 k
x

2. @The answer should be obvious just by 8 @Note that

looking at it. Testing n 0 gives us:
— 36
which happens when x — -3. Then
= (4)2
x2 (-3)2 = 9

16 — 16

We could also expand and solve like so: 1

16 n + 8 n —E 16 -1)

16n —o fi

m
10. Multiply both sides by 2 to get 4,
n
n 1
which means Then,
b
1
ac n 1

b ac
2m 24 8

11. If x2 — 12 — 4, then x2 16 and x — +4.

If b ac, then b — ac must equal 0. Since x < 0, x
I@lf3x 8 —23, then 3x -15. If x2 +7— 21, then x2 — 14 and
Multiplying both sides by 2, 6x —30 and 3 3 17.
¯ 37.
13. Because this is a no calculator question,
5' @Cross multiply. guess and check is a valid strategy. You can
also do the following:
4 8
9 3
x2(x4 — 9) — 8x
4

12 72m x —9x — 8x4

1
m - 8x2 - 9) —
6
x2(x2 — + 1)
x2(x _4_ 3) (x — + 1) —o
—
-8
—9 Because x > 0, x must be 3 for the equation
above to be true.

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14. 18. @
6 d
3

18 a
2

1
— 81 2 24
x — 77
12

11
15.

19.
10
- 2y)
3x — 6y
3x — 6y + 3z
6
x — 2y + z
36

20.
16.

8 + 5x xy2+x Y2 -1—0
8 + 5x 2x 10
-18
—6
Since Y2 + 1 is always positive, x must equal
1.
17. Cross multiply.
21.
X X 1.2

6 42 ITZ2g — 111711 g
42 x— 6x + 72 ITZI + 1112

36x — 72 a(ml + 1712) — nt2g pnqg

a(ml + 1712) —

6 6
IT12g a(ml + m2)
Now, nqg
x 2

22. @Because the 2's cancel out, the

acceleration stays the same.

2m2g y (2m1)g 2(m2g — HTT11g)

anew
+ 2m'2 nt2)

aold

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CHAPTER 28 ANSWERS TO THE EXERCISES

23. @Divide both sides by P and take the tth

root of both sides.

24. @From the previous question, we know

that r — 1 Because V is half P,

Thus, r 13

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Chapter 8: More Equation Solving Strategies

CHAPTER EXERCISE:

1. @Expand the right side.

(2x + 3) (ax 12x2 bx - 15

2ax2 + 3ax 10x ¯ 15 -15

2ax2 + (3a — 10)x — 15 — 12x2 bx 15

Comparing both sides, 2a — 12 and b 3a 10, which yields a 6 and b 10 3(6) — 10 8.

2• @Expand the left side:

x2 + 9y2 + 42

+ 6xy 9M + _4-42

¯ 42

22 — 49

3. Multiply both sides by b.

a + 5b
5b

4. 2 Multiply both sides by x (x

(x x(x — 4)

—4 X 4x

We can see that x 2.

5. @Expanding the right side,

4x2 m x —h + + n2
Comparing both sides, we see that
9 n and m 4n

Therefore, n —3 and m 4(—3) — -12

-12 + (—3) — -15

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6. Multiply both sides by xyp.

1 1 1

yp+xp —xy
yp — xy — xp
yp x(y p)

6x2 -b cx 7

6x2 + 3ax + 2bx + ab

6x2 -b cx 7

Comparing both sides, 3a + 2b c and ab 7. The two possibilities are a — 1, b 7 and a — 1.

Therefore, c 3a + 2b 3 (I) 17 and c 3a +2b 3(7) +2(1) 23.

8. C Multiply both sides by 2x + I.

mx 23 — I) 41

+ mX 23 12x2 + 6x — 36x 18 41

—I— x 23 — 12x2 - 30x + 23

Comparing both sides, m— —30.

9. @Expand the left side of the equation.

( x-•3 + k x2 — 3) (x — 2) kx3 — 3x — 2x3 — 2kx2 + 6

— x4 + (k 2)x3 2kx2 - 3x + 6

Comparing this to x4 + 7x3 — 18x2 — 3x + 6, we can see that k -2 7 and -2k —18. In both cases,

10. @Multiply both sides by (n — l)(n + 1).

-1) —1)
311 3+2n 2 2n — 3012 - 1)
2
n 3 -3

n — 3 or — 2. Because n > 0, n — 3.

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11. 4 or 6 Multiply both sides by (x + 2) (x 2).

12(x -2) + 2) -2)

12x 24—2x -4 x 4

lox - 28 4
2
0 x ¯ I Ox 24
0

So the two possible solutions are 4 and 6.

11
12. Notice that x 2 (x + l)(x 1) on the right hand side. It's then easy to see that we should
2

multiply both sides by (x + 1) (x 1)

= 35
- 35

35
6x — 33
11
2

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Chapter 9: Systems of Equations

CHAPTER EXERCISE:

1. @Substituting the second equation into the first,

-11
5y — —11
3 —11
14 y — 14
y 1

Finally, x I — 3(1) — —2.

2. From the first equation, y — 20 2x. Plugging this into the second equation,

6x -5(20 - 2x) — 12
6x - 100 + lox — 12

16x 112

We already know the answer is (D) at this point, but just in case, y 20 2(7) 6.

3. B Add the two equations to get 7x — 7y 35. Dividing both sides by 7, x —y 5. We can multiply
both sides by —1 to get y — x —5.

4. C The fastest way to do this problem is to subtract the second equation from the first, which yields

5. In the first equation, we can move 3x to the right hand side to get y — —5x + 8. Substituting this into
the second equation,

-10
16 -10
-131 -26

Then, y — -5(2) +8 —2. Finally, xy (2) (Q) —

6. The two graphs do not intersect at all, so there are no solutions.

7. [0] From the first equation, we can isolate y to get y —5x 2. Substituting this into the second equation,

2(2x - 1) = - 2)

3 15X 6
15x +9
-11 llx

Finally, y —

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8. @Divide the first equation by 2 to get x — 2y 4. We can't get the coefficients to match (—2 vs. 2 for
the y's). Therefore, the system has one solution. In fact, we can even solve this system by adding the two
equations to get 2x 8, x 4, which makes y 0.

9. @To get the same coefficients, multiply the first equation by —2 to get —4x + I()y — —2a. Now we can
see that —2a —8, a 4.

10. @Multiply the first equation by —3 to get —3ax 6y —15. The constant a cannot be —1. Otherwise,
the second equation's coefficients would then be equal to the first equation's coefficients, resulting in a
system with no solution.

11. @First, multiply the first equation by 3 to get rid of the fraction: 12x —y —24. Next, substitute the
second equation into the first,

— + 16) ¯ ¯ 24
12x - 4x 16 -24
8x —8

Finally, y 1) +16— 12.

12. 10 We can isolate x in the second equation to get x y — 18. Substituting this into the first equation,

Y — o.5(y — 18) + 14

Y o.5y _ 9 14

o.5y 5
Y 10

13. match the coefficients, multiply the first equation by 18 to get 6x — 3y == 72. We can then see that
a 3 ifthe system is to have no solution.

14. Divide the first equation by 3 to get x — 2y == 5. Divide the second equation by —2 to get x — 2y 5.

They're the same, so there are an infinite number of solutions.

15. For a system to have infinitely many solutions, the equations must essentially be the same. Looking
at the constants, we can make them match by multiplying the second equation by 2. The equations then
look like this:

mx —
6y 10

4x 2ny — 10

m 4
Now it's easy to see that m — 4 and 2n 6, n 3. Finally,
n

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CHAPTER 28 ANSWERS TO THE EXERCISES

16. Plugging the first equation into the second,

—3
—3
6
X — 36

Therefore, y — 36+3 9.

17. Let s, m, and I be the weights of small, medium, and large jars, respectively. Based on the information,
we can create the following two equations:

16s 2m I

To get the weight of the large jar in terms of the weight of the small jar, we need to get rid of m, the weight
of the medium jar. We could certainly use elimination, but here, we'll use substitution. Isolating m in the
second equation, m = I — 4s. Substituting this into the first equation, we get

16s = — 4s) + 1
16s 21 — 8s + I
24s 31

Eight small jars are needed to match the weight of one large jar.

18. D Since there were 30 questions, James must have had 30 answers, x + y 30. The points he earned
from correct answers total 5x. The points he lost from incorrect answers total 2y. Therefore,5x 2y 59.

19. 5 Let a and b be the number points you get for hitting regions A and B, respectively. From the
information, we can form the following two equations:

a +2b — 18

2a +b 21

To solve for b, multiply the first equation by 2 and subtract to get 3b 15, b 5.

20. Let r and c be the number and circular tables,

of rectangular tables respectively, at the restaurant.
Based on the information, we can make the following two equations:

4r + 8c _ 144
r + c 30

To solve for r, multiply the second equation by 8 and subtract to get —4r — -96, r — 24.

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Chapter 10: Inequalities

CHAPTER EXERCISE:

X ¯4 ¯ 14

-5x > -10

Of the answer choices, only —1 is a solution.

2. Multiply both sides by 4 to get rid of the fractions.

3
—x —4> —x — 10
4
3x - 16 > 2x - 40

3. C The shaded region falls below the horizontal line y 3, soy < 3. The shaded region also stays above
y — x, soy > x.

4. @Let's say Jerry's estimate, m, is 100 marbles. If the actual number of marbles is within 10 of that
estimate, then the actual number must be at least 90 and at most 110. Using variables, m — 10 n <

5. @Setting up the inequality,

12P + 100 > -31) + 970

15P > 870
P> 58

3n — 6 > —4n + 36
711 > 42

Since n is an integer, the least possible value of n is 7.

7. @The shaded region is below the horizontal line y = 3 but above the horizontal line y -3. Therefore,
y —3 and y 3.

16
8. The time Harry spends on the bus is — hours and the time he spends on the train is — hours. Since

8 16
the total number of hours is never greater than 1, —+ — < 1.

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CHAPTER 28 ANSWERS TO THE EXERCISES

9. If the distributor contracts out to Company A for x hours, then it contracts out to Company B for
10 — x hours. Company A then produces 80x cartons and Company B produces 140(10 — x) cartons.
Setting up the inequality,
80x + 140(10 - x) > 1, 100

3
10. @The line going from the bottom-left to the top-right must be y —x
2
+ 2 and the line going from the
top-left to the bottom-right must be y —2x + 5 (based on the slopes and y-intercepts). Answer (D)
3
correctly shades in the region above y — —x + 2 and below y —
2

11. @Plug in x — 1, y — 20 into the first inequality to get 20 > 15 + a, 5 > a. Do the same for the second
inequality to get 20 < 5 b, 15 < b. So, a is less than 5 and b is greater than 15. The difference between
the two must be more than 15 — 5 — 10. Among the answer choices, 12 is the only one that is greater than
10.

12. One manicure takes 1/3 of an hour. One pedicure takes 1/2 an hour. The total number of hours she

spends doing manicures and pedicures must be less than or equal to 30, so —m + —p 30. She earns

25m for the manicures and 40p for the pedicures. Altogether, 25m + 40p 2 900.

13. @From the given inequality, x 3k + 12. Subtracting 12 from both sides gives x — 12 3k, which
confirms that I is always true.
From the given inequality, 3k + 12 k, which means 2k > —12, k > —6, so II must also be true.
From the given inequality, k x. Subtracting k from both sides gives 0 x — k. Therefore, Ill must also
be true.

9 10
14.
4
< x < — Let's solve these separately. First,

20
3

-20 < -6x + 12

-32 < —6x
16
3

Now for the second part,

9
2

-4x < -17

17
4

17 16 10
Putting the two results together, — < x < —. Therefore, — < x — 2 < —.
3

15. the area is at least 300, then xy 300. The perimeter of the rectangular garden is 2x + 2y, so
2x + 2y 70, which reduces to x +y 35.

16. C I is not always true because of negative values. Take a — —5 and b 2 for example. a < b, but
a > b2. II is definitely true. It's the equivalent of multiplying both sides by 2. Ill is also true. It's the
equivalent of multiplying both sides by —1, which necessitates a sign change.

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Chapter 11: Word Problems

CHAPTER EXERCISE:

1. [01 The square of the sum of x and y is 7. @Converting 75% and 85% to fractions in
(x + y)2. The product of x and y is xy. The the equation below,
question asks for the difference: (x + xy
3 17
-(68) n
4 20
17
98 x - 10) 51
20
98 X — 3X - 30
—128
x — 32
60 n

3. 18

15 +N 2
4
x —16 Cross multiplying,

18
+N 15

8 + 2N — 15 + N
4. 5 or 10 Based on the information, we can
form the equation 4x + 10y 60. Now it's
just a matter of guess and check. Since x and
y are integers, it won't be long before we find 9. They start with the same number x. Once
something that works. For example, Alice gives 16 to Julie, Alice is left with x — 16
x 5, y 4 is one possible solution. and Julie then has x + 16.

5. @The store's monthly total cost is X 16 16)

3, 000 + 2, 500x. For an entire year, we
X -4—16 2x - 32
multiply by 12 months:
— 12(3, ooo + 2,500x). x
x _ 48
6. @Susie bought 2x pounds of salmon and y
pounds of trout. The total cost is then
10. C The fraction of students who take math is
(3.50) (2x) +5y 77 1 1 1 11
1 Let x be the total
7x + 5y — 77
number of students.
Since x and y must be integers, we can plug
each answer choice into the equation above to 11
x — 33
see if an integer value for x. When
we get 24
y 4, for example, x 8.14, which is not an x—72
integer. The answer turns out to be 7. When

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11. Let x be the number of trades. Each trade, 15. Let x be the number of seashells that Carl
Ian has a net gain of 1 card while Jason has a
net loss of 1 card.
has. Bob then has —x seashells and Alex has
3
20 -b ¯ 44
x x —x seashells.
2
Since Alex and Bob together
2x — 24 have 60 seashells,

x — 12
1
— 60
2
12. Making an equation to figure out x, 2x — 60
21
x — 30
24 Carl has 30 seashells.

12

13. Three times the price of a shirt is 120.

Since atie, which costs 30, is k less than that, k

must be 120 — 30 90. As an equation,

tie — 3(shirt) —k
30 3(40) - k

30 120 -k
k — 90

14. D number of $5 coupons given out

Let the
be Then the number of $3 coupons given
x.

out is 3x, and the number of $1 coupons

given out is 2(3x) 6x.

5(x) +3(3x) + I(6x) 360

5x 9x + 6x 360
20x 360
x — 18

The number of $3 coupons given out is then

3x = 3(18) — 54.

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Chapter 12: Lines

CHAPTER EXERCISE:

1. A vertical line that intersects the x-axis at 5. the two lines intersect at the point (2, 8),

3 has an equation of x = 3. then both lines must pass through that point.
Plugging the point into the equation of the
second line, we can solve for b,

1 —bx
3 8
1

1 Plugging the point into the equation of the

first line,
6
2

12 — 2a

6 a
3 1
3' The slope of line I is
9 3
6.
Using point-slope form,
Y2 Yl 2 1

m(x — Xi) —4 2

1
7. @From the graph, slope m is positive and
y-intercept b is negative. Therefore, mb < 0.
1

3 @The line y — —2x — 2 has a slope of —2

and a y-intercept of —2. Line must have a
I

At this point, we test each answer choice by

slope that is the negative reciprocal of —2,
plugging in the x-coordinate and verifying
the y-coordinate. Only answer (A) works. which is —. Since they have the same

y-intercept, the equation of line I must be

4. C The graph of line I goes up three units for
1
every two units to the right, which means its
2
slope is . A parallel line must have the same
slope.Only answer choice (C) gives an
equation of a line with the same slope.

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CHAPTER 28 ANSWERS TO THE EXERCISES

9. @The line we're looking for must have a

1
slope that is the negative reciprocal of
2'
which is —2.

Plugging in the point (1, 5),

Now that we have b, the line is y —2x + 7.

10. @
10 —4 2
3
6 2
3

Cross multiplying,

— 18

2x -2 — 18

20
x — 10

11. f is a line, then f must be a flat line.

That's the only way that both f (2) f (3)
and f (4) f (5) can be true. Since f is flat
and f (6) = 10, then all values of f are 10, no
matter what the value of x is. Therefore,

f(o) 10.

12. One easy way to approach this problem is

to make up numbers for a and b. Let a 1
a 1
and b = 2 so that Since the second line

d
is perpendicular to the first, —2, which

satisfies the condition in answer choice (A).

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Chapter 13: Interpreting Linear Models

CHAPTER EXERCISE:

1. @The slope is —3, which means the water level decreases by 3 feet each day.
2. B The value 18 refers to the slope of —18, which means the number of loaves remaining decreases by 18
each hour. This implies that the bakery sells 18 loaves each hour.

3. @The y-intercept of 500 means that when n 0 (when there were no videos on the site), there were 500
members.

4. @The number 2 refers to the slope of —2, which means two fewer teaspoons of sugar should be added
for every teaspoon of honey already in the beverage. Don't be fooled by answers (C) and (D), which
"reverse" the x and the y (h and s, in this case). The slope is always the change in y for each unit increase
in x, not the other way around.

5. The salesperson earns a commission, but on what? The amount of money he or she brings in. To get
that,we must multiply the number of cars sold by the average price of each car. Since x and c already
represent the commission rate and the number of cars sold, respectively, the number 2,000 must represent
the average price of each car.

6. @The number 2,000 refers to the slope, which means a town's estimated population increases by 2,000
for each additional school in the town.

7. @The number 4 refers to the slope of —4, which means an increase of 10 C decreases the number of
hours until a gallon of milk goes sour by 4. In other words, the milk goes sour 4 hours faster.

8. @When — t 0, there is no time left in the auction. The auction has finished. Therefore, the 900 is the
final auction price of the lamp.

9. @Because it's the slope, the 1.30 can be thought of as the exchange rate, converting U.S. dollars into
euros. But after the conversion, 1.50 is subtracted away, which means you get 1.50 euros less than you
should have. Therefore, the best interpretation of the 1.50 y-intercept is a 1.50 euro fee the bank charges
to do the conversion.
2
10. see the answer more clearly, we can put the equation into y = mx + b form: — —x t -l- The slope
5

is , or 0.4, which means the load time increases by 0.4 seconds for each image on the web page.

11. @The slope is the change in y (daily profit) for each unit change in x (cakes sold).
12. @Notice that the y-intercept is negative. It is the bakery's profit when no cakes are sold. Therefore,
anything that varies with the number of cakes sold is incorrect. For example, answer (D) is wrong because
the cost of the cakes that didn't sell depends on how many the bakery did sell. It's not a fixed number
like the y-intercept is. The best interpretation of the y-intercept is the cost of running the bakery (rent,
labor, machinery, etc.), which is likely a fixed number.

13. @The solution means that the bakery's daily profit is zero when 5 cakes
(5, 0) are sold. Therefore,
selling five cakes is enough to break-even with daily expenses.

14. The slope of the equation is 5, which means the temperature goes up by 5 degrees every hour. So
every half hour (30 minutes), the temperature goes up by 0.5 x 5 — 2.5 degrees.

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1
15. @Putting the equation into y mx + b form, y — —x + 7. The slope of means that one more turtle
2
requires an additional half a gallon of water. So Ill is true.

Getting x in terms of y, x 2y 14. The "slope" of 2 means that 1 more gallon of water can support two
more turtles. So I is true.
16. C Because this question is asking for the change in "x" per change in "y" (the reverse of slope), we need
to rearrange the equation to get x in terms of C.

c— 1.5 + 2.5x

Dividing each element in the equation by 2.5,

o.4C 0.6+ x

x 0.4C — 0.6
The slope here is 0.4, which means the weight of a shipment increases by 0.4 pounds per dollar increase
in the mailing cost. So a 10 dollar increase in the mailing cost is equivalent to a weight increase of
10 x 0.4 4 pounds.

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Chapter 14: Functions

CHAPTER EXERCISE:

1. @Check each answer choice to see whether f (0) 21, and f (3) = 29. The only function
that satisfies all three is (D).

2. [01 f (x) g(x) when the two graphs intersect. They intersect at 3 points, so there must be 3 values of x
where f (x) g(x).

3. —2. Now where else is f at —2? When x —3. So a must be —3.

4. @Draw a horizontal line at y — 3. This line intersects f (x) four times, so there are four solutions (four
values of x for which f (x) 3).

5. @Plug in —3 and 3 into each of the answer choices to see whether you get the same value. If you're
smart about it, you'll realize that answer (C) has an x2, which always gives a positive value. Testing (C)
out, f (—3) = 3(—3)2 + 1 — 28 and f (3) 3(3)2 + 1 28. The answer is indeed (C).

@First, g(10) = f(20) - 1. 3(20) +2 62. Finally, g(10) 1 61.

32
—8

8. C We plug in values to solve for a and b. Plugging in (0, —2), —2 +b b. So, b —2. Plugging
in (1, 3),

So a 5 and f (x) 5x2 — 2. Finally, f (3) 5(3)2 — 2 = 43.

1
9. Plug in the answer choices and check. f , which is less than —. The answer is (A).

10. @The x-intercepts of —3 and 2 mean that f (x) must have factors of (x + 3) and (x 2). That eliminates
(C) and (D). A y-intercept of 12 means that when we plugin x 0, f (x) 12. Only answer (B) meets all
these conditions.

11. [Og(2) —1 = 3. So, f(g(2)) f(3) 32-1- 1 — 10.

12. @Draw a horizontal line at y c, passing through (0, c). This horizontal line intersects with f three
times. That means there are 3 values of x for which f (x) = c.
13. @
2f(k) —8
f(k)

Looking at the chart, f (x) — 4 only when x — 3. Sok = 3.

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CHAPTER 28 ANSWERS TO THE EXERCISES

14. 18 2 16 4. f (II) 11 —2 — — 3. f (18) f (11) 4 —3 — 1. Testing each

answer choice, f (3) is the only one that also equals 1.

15. (1, 2) cannot be on the graph of y since an x-value of 1 would result in division by 0.

16. D

3a
3a — 36

12

17. Using the table, g(—l) 2. Then, f (2) 6.

18' g(c) 5, then c 1 since 1 is the only input that gives an output of 5. Then, f (c) — 3.

19. B From the second equation, f (a) 20. So,

—3a +5
20
3a — -15
a —5

20. = — I) = f (5) 2. We get f (5) 2 from the table.

21. f (8) 4(8) — 3 29. Testing each answer choice to see which one yields 29, we see that g(8) —
3(8) +5 29.

22. @When x 0, y 9, so the y-intercept is 9. When y 0, x 3, so the x-intercept is 3.

x
o 3

A AOB is a right triangle with a base of 3 and a height of 9. Using the pythagorean theorem,
A02 + 0B2 AB2

92 + 32 AB2

90 AB2

3 10

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23. The graph of g is 4 units up from where f is, but because the slope of f is —2, the x and y intercepts
of g will not increase by the same amount. They'll increase in a ratio of 2:1. So when the y-intercept gets
shifted up by 4, the x-intercept gets shifted to the right by 2. The new x-intercept is therefore 1 + 2 3.

Another way to do this is to actually solve for the x-intercept. Using slope-intercept form, we get f (x) —
—2x 2. Adding 4 to get the equation of g, g(x) — —2x + 6. Setting g(x) 0 and solving for x to get the
x-intercept, we get x 3.

24. The function g(x) is a line with a slope of 1 and a y-intercept of k. If you draw g(x) with the different
possibilities for kfrom the answer choices, you'll see that there's an intersection of 3 points with f (x)
only when k — 1 as shown below.

x
2 1 o 2

25. @Plugging in ( —a, a),

a 12

12

- 12 —o

Since a > 0, a

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CHAPTER 28 ANSWERS 10 THE EXERCISES

Chapter 15: Quadratics

CHAPTER EXERCISE:

1' factor to find the x-intercepts.

y = -3x 10

The x-intercepts are 5 and —2. The distance between them is 5 — (—2) 7.

2. @Using the quadratic formula,

x
2(1) 2 2

3. .5

If you had trouble factoring this, remember that you can always use the quadratic formula. Since a < 1,

1
or 0.5.
2'

4. D Move the 8 to the left side to get 3x2 + 10x 8 0. Now, we can either use the quadratic formula or
factor. In this case, we'll go with factoring.

2
So, x — 4 or x — —. Since a > b, b must be —4 and b2 — 16.
3

5. @Expanding everything,

4x 12x +9
4x 16x + 4 —o

b —16
The sum of the solutions is
a 4

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6. @Substituting the first equation into the second,

3 —x +cx

The system of equations will have two solutions if the equation above has two solutions. For the equation
above to have two solutions, the discriminant, b2 — 4ac, must be positive.

2
c 12 > o

c2 > 12

Testing each of the answer choices, only answer (A), —4, gives a value bigger than 12 when squared.

7. find the intersection points, treat the two equations as a system of equations. Substituting the first

equation into the second,

4 5

9
+3

The y-coordinates of the intersection points must be 4 (from the first equation), so the two points of
intersection are (—5, 4) and (1, 4).

8. C Because the vertex is at (3, —8), the answer must be either (A) or (C). Because the parabola passes
through (1, 0), we can use that point to test out our two potential answers. When we plug in x 1 into

(C), we get y — 0, confirming that the answer is (C).

9. 2.5 From the equation v 5t — t2 —

t(5 — t), we can see that the t-intercepts are 0 and 5. Because the
maximum occurs at the vertex, whose t-coordinate is the average of the two t-intercepts, t 2.5 results
in the maximum value of v. You can confirm this by graphing the equation on your calculator.

10. find the minimum number of mattresses the company must sell so that it doesn't lose money, set

m2 100m — 120, OOO — O

(m - 400) (m + 300) = O
m —300,400

Since it doesn't make sense for the number of mattresses sold to be negative, m 400. If you had trouble
factoring the equation above (it's tough), the graphing calculator and the quadratic forrnula are both
good alternatives.

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CHAPTER 28 ANSWERS TO THE EXERCISES

11. Substitute the first equation into the second,

—3 —ax +4x—4

For the system to have one real solution, the equation above should have only one real solution. In other
words, the discriminant, b2 — 4ac, must equal 0.

(4)2 — 40) —o
¯
16 + —0
—16
a —4

12. B We need to complete the square. First divide everything by —1,

—y x2 — 6x 20

Now divide the middle term by 2 to get —3 and square that result to get 9. We put the —3 inside the
parentheses with x and subtract the 9 at the end.

2
— 20 — 9

Now simplify and multiply everything back by —1.

(x - +29

13. One of the x-intercepts is 3. Since the x-coordinate of the vertex, 5, must lie at the midpoint of the
two x-intercepts, the other x-intercept is 7. Therefore, k 7, giving us y a(x — 3) (x — 7). We can now
plug in the vertex as a point to solve for a.

-32 7)

-32
--32 4a

a 8

14. @Substituting the point (3, k) into both equations,

k 2(3) +b
k — (3)2 +3b +5

This is a system of equations. Substituting the first equation into the second,

2(3) 4b (3)2 3b +5

— 3b + 14
8 — 2b
b —4

From the first equation, k

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Chapter 16: Synthetic Division

CHAPTER EXERCISE:

5. z 1 is a factor only ifthe polynomial

yields 0 when z = 1 (the remainder theorem).
4 Therefore, we can set up an equation.

8 —o
8 0

8 From here, we can see that k 7.

This result can be expressed as 4 +
6.

x 2
1
3x2 4
1 6x2 2
3x2

6x 4
_l_ 2
6x

8
1
This result can be expressed as
This result can be expressed as 8
from which A x 2.
1
from which Q
7. @This question is asking you to divide the
3. 6 This question is asking you to divide the expression by x + 1 and write the result in the
expression by 2x — 1 and write the result in form of
the form of Dividend — Quotient x Divisor + Remainder.
Dividend — Quotient x Divisor + Remainder.
6
2x + 1
x 1 2x2 3
2x-1 4x2 + 5
2x2
4x2
6x 3
_l_ 5
6x 6
1
3
6
Therefore, 2x2 3 — (2x
Therefore, 4x2 +5= (2x + 1) (2x

4. @Using the remainder theorem, the

remainder when g(x) is divided by x + 3 is
equal to g(—3) = 2.

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CHAPTER 28 ANSWERS TO THE EXERCISES

8. @This question is asking you to divide the 13. A From the remainder theorem, 3x 1 must
expression by x 2 and write the result in the
form of be a factor of p(x) if p
Dividend Quotient x Divisor + Remainder,
where ax + b is the quotient and c is the
remainder.

x2 — 2x

6x 9

6x 12

Therefore, x2 + 4x 9 (x + 6) (x — 2) + 3.
Finally, a — 1, b 6, c 3, and a + b + c — 10.

9. C Using the remainder theorem, p(2) =0

means that x — 2 is a factor of p(x).

10. C Use the remainder theorem to test each

option for a remainder of 0.

p(2) 23 +22-5(2) +3 5.
p(l) — 1 +12 — 5(1) +3 0.

p(—3) 5(—3) +3 — 0.
Therefore, p(x) is divisible by x — I and x + 3.

11. If p(x) is divisible by x — 2, then p(2)

must equal 0 (the remainder theorem).
Testing each answer choice, only choice (D)
results in 0 when x 2.

12. C Using the remainder theorem, we can set

up a system of equations. When the
polynomial is divided by x — 1 or x + 1, the
remainder is 0, which means that if we let
p(x) denote the polynomial, p(l) 0 and
p(—l) 0.

+ _ + 5(1) —o

a b -3—5 —o
Adding the equations together,

2a -6=0

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Chapter 17: Complex Numbers

CHAPTER EXERCISE:

1.

—5-3i+2 5i -7 -8i

2.

i4 + 3i2 + 2

(6 + 20 (2 + 5i) — 12 + + 4i + 12 ¯ 2 + 34i
Therefore, a —

+2) -2(5 10

3i(i 2) — — 1) —

7.

(i4)23 i = (1)23

8.

32 6i

(5 - - 3i) 20 - 15i - 8i + 612 20 23i - 6 14 - 23i

10.

1 1 1 1 1
—1+1
Multiplying both top and bottom by i,

1
—i

11.

(3
3 — i — 9i + 312 3 IOi + 31 •2
3 IOi —3 -IOi
9 9 — i2 9 1 10

12.

22 2i — 2i+i 2 4 4 3 4
4 5 5 5

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 18: Absolute Value

CHAPTER EXERCISE:

3(1) I ¯ 41 4

2. Only the expression in answer (B) can equal —5 (when x 1 or 3). Because the absolute value of
anything is always greater than or equal to 0, the other answer choices can never reach —5.

3. Recall that the graph of y IXI is a V-shape centered at the origin. The graph pictured is also
V-shaped but converges at y 2, which means ithas shifted two units down. Therefore, the equation
of the graph is y — xl — 2. Note that y — lx — 21 shifts the graph two units to the right, NOT two units
down.
4. D Test each of the answer choices, making sure to include the negative possibilities. For example, the
answer is not (A) because when x 2 or — 2, x — 31 is not greater than 10. However, lx — 31 is greater
than 10 when x -8.

5. Smart trial and error is the fastest way to find the bounds for x. The lower bound for x is —8 and the
upper bound is —4. There are 5 integers between —4 and —8 (inclusive). If we wanted to do this problem
more mathematically, we could set up the following equation:

Subtracting 6,
-3
Since x is an integer,

6. n is positive,

n —2 — 10
n — 12

If n is negative,

n 2 -10
—8

The sum of these two possible values of n is 12 + (—8) —

7. the graph of If (X) l,all points with negative y-values (below the x-axis) are flipped across the

x-axis. All points with positive y-values stay the same. Graph (D) is the one that shows this correctly.

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8. Let's make up a number, x 3. Then b 13-101 7, and b x — 7Using our numbers,

3 4.

we're looking for an answer choice that gives 4 when b 7. (C) and (E) are the only choices that do

that. Now let's make up another number, x 8. Then b 18 — 101 = 2, and b — x = 2 — 8 -6. The
only answer choice that gives us —6 when b 2 is (C). If we had chosen numbers that didn't narrow our
choices down, then we would have kept guessing and checking. If you use this strategy wisely, however,
you'll never have to guess for too long.

To do this question mathematically, we have to realize that when x < 10, x 10 is always negative.
Therefore,

x -10 —b
x —10 b

Using substitution, b x becomes b (10 b) = 2b - 10.

9. C The midpoint of 6— and 6— is the average: 6 +6 /2 6—. The midpoint is — away from the

boundaries of the accepted range for the length of a hot dog. So whatever h is, it must be within of the

midpoint:
1

10. D The midpoint of 400 and 410 is the average: (400 + 410)/ 2 The midpoint is 5 away from the
405.
boundaries of the accepted range for the length of a roll of tape. So whatever I is, it must be within 5 of
the midpoint:
II -4051 < 5

11. B There are two possible values of x, 3 and —1. There are two possible values of y, 1 and —5. We get
the smallest possible value of xy when x 3 and y — —5, in which case xy -15.

12. C If al < 1, then by definition,

This means that Ill is true. Because a must be a fraction, a2 < I, so II is also true. However, I is not always

true because when a is negative, — is not greater than 1.

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Chapter 19: Angles

CHAPTER EXERCISE:

I. Using the exterior angle theorem, 5. Because alternate interior angles are
equal, one of the missing angles of the lower
triangle is also a:

140 50 +j
90 =j
1

2. The missing angle in the left triangle is

1800 — 600 500 700. This angle is an
exterior angle to the triangle on the right. So, x
using the exterior angle theorem,
a
m
70=y+40
30 Y

Since x is an exterior angle to the lower

3. B a + b + c + d is equal to the sum of the
triangle,
angles of the quadrilateral, as shown below.

6. @The angle at the top of the triangle is

m 180 — 70 — 30 80. If we look at the larger
triangle, taking away the top angle gives

a+b — 180 - 80 100

do

do
7. C The angles form a circle, which means
theysum to 3600.

Because the angles of a quadrilateral sum to x + y ¯ 3600 — 450 800 2350

360, the answer is 360.

8. @Filling out the bottom triangle, the

missing angle is 1800 - 600 400 — 800,
— 180 which means the angle across from it in the

40 +y+ (40 + y) — 180 upper triangle is also 800. Finally,

2y + 80 180
z 1800 - 450 - 800 550
2y = 100

y = 50 9. @The two angles form a line, which means

they sum to 1800.

180

2x + 40 — 180

140
x — 70

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10. @We can figure out two angles within the 15• @Angle a is equal to 180 -60 — 120.
triangle: 1000 and 500. Because y is an Angle b is equal to 180 40 — 140. Finally,
exterior angle, we can use the exterior angle a —I— b — 120 —I— 140 — 260.
theorem to get its value:

y 100+50 150 400

11. c
600
m
Shaded Angles Angles of Rectangle
+ Angles of Quadrilateral 600 400 b
n
— 360 + 360
720

12. @The angles of any polygon sum to

180(n — 2), where n is the number of sides.
The angles of a hexagon (6 sides) sum to
180 (6 2) Because the hexagon is
720.
regular, all angles have the same measure.
Therefore, each angle is 720 6 — 1200.
:

Finally,
x 120 - 90 30

13. @bis an alternate interior angle to the 450

angle, which means they're equal: b — 45. a
and c are also alternate interior angles so
180 — 45 — 135. Using these values,
we can see that all three are true.
14. @The two missing angles in the smaller
triangleadd up to 800. The two bottom
angles in the larger triangle add up to
180 — 70 — 110. If we take the two missing
angles of the smaller triangle away from the
two bottom angles of the larger triangle, we'll
end up with x + y.

x+y— 110 80=30

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Chapter 20: Triangles

CHAPTER EXERCISE:

1. @Because the hypotenuse is always the

largest side, x + 5 must be the hypotenuse
while x and x 2 must be the legs. Using the
pythagorean theorem,
5 5
3
x2 + (x —

2. @Using the 30 — 60 — 90 triangle M 4 4

1 1 Drawing the height splits the base into two

relationship, DC -BC 10) -
2 equal parts of length 4. From the 3 4 — 5

3. @Using the 45 — 45 — 90 triangle pythagorean triple, we know the height is 3.

relationship, x 61/2.
The area is then — (8) (3) — 12.

4. 8 Triangles ABE and DCE are similar. 9. @The side length of the square is — 2.
Therefore, Draw the height of the triangle to create two
30 — 60 — 90 triangles:
CD
CE BE
6
3 4

8
1 1

5• TO satisfy the triangle inequality

theorem (any two sides of a triangle must
sum up to be greater than the third side), the
third side must be less than 3 + 13 16 and The area of the triangle is then
greater than 13 —3— 10. 1

6. 55 If two angles have the same measure,

then the sides opposite them have the same
length. To get the largest perimeter, we
choose the third side to be 20. The perimeter
is then 15 + 20 + 20 55.

7. is invalid because 6 +7< 14. II is

invalid because 5 +5< 12. Ill is the only one

that satisfies the triangle inequality theorem
(any two sides of a triangle must sum up to
be greater than the third side).

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10. @Let the height of the bottom piece be x. 13.

The height of the cone and the radii of the 57T

2250 x
circles forrn two similar triangles as shown 1800 4
below.

14. @The sides of triangle DEF are 9 : 6 — 1.5

times longer than the respective sides of

1
triangle ABC. Therefore, EF =9x 1.5 — 13.5
and DF = 5 x 1.5 = 7.5. The perimeter of
triangle DEF is then 9 + 13.5 + 7.5 = 30.
x
15. @Because of the triangle inequality
theorem, the third side must be less than
6
8 +20 28 but greater than 20 - 8 — 12. so
Using the similarity, 12 < p < 28. There are 15 integers from 13 to
27.
1

2 6
16. @We can use the pythagorean theorem to
find BC:

Cross multiplying,
AC2 + A 132 — BC2
122 + 92 — BC2

225 BC2
15 BC
11. @To satisfy the triangle inequality theorem
(any two sides of a triangle must sum up to Note that this is a multiple of the 3 -4-5
be greater than the third side), x + y must be triangle.
greater than whatever z is. z cannot be 8,
because then x + y would add up to just 2. In c
fact, z cannot be 5, 6, 7, or 8 because in all

those cases, x+ y does not exceed the value

The greatest possible value of z is 4 (x
of z.
and y could both be 3). 15
12

12. 2.5 Triangles GEF and GHC are similar.

6
Solving for EF,

HC
9

10 Now ACDE is similar to ACAB.

2 5
CE CB
DE
Triangles ADF and GEF are also similar. So, CE 15
6 9
AD
Cross multiplying,
AD 2
5 4 9(CE)
5 CE —
AD — 2.5 10
2

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CHAPTER 28 ANSWERS TO THE EXERCISES

17. If BC is the shortest side in the isosceles 20. Using the pythagorean theorem,
triangle, then AB AC and ZA is the
smallest angle. At the same time, we want to 82 + x2
maximize ZA so that Z B is minimized. Now
if all the angles were 600, then the triangle 64 + x2 x2 + 4x + 4
would be equilateral and BC wouldn't be the 64
shortest side. So we need to decrease ZA to
the next highest option, 500 , which minimizes
15 x
ZB to 130 + 2 650.

21. Label what you know.

650

1
2

650 500 2
c 1

18. B Draw the extra lines shown below and AIZ 2

use the 8 — 15 — 17 right triangle. All triangles in the diagram are 45 — 45 — 90,
which means W Z — XY and
WX zy — 2v/ä. The perimeter of WXYZ is
17
8 6Vä.
22. @From the coordinates, AB — 7 and
z
15 BC Because ZABC is a right angle,
7.
20
triangleABC is a 45 — 45 90 triangle.
Therefore, the measure of ZBAC 450
12
which is 450 x — radians.
1800 4

x 15 23. The smaller triangle in the first quadrant

is a3 —4 5 triangle and is similar to triangle
19. C Draw an extra line to complete the AOB. Using the similarity,
rectangle. Then use the 7 — 24 25 right
triangle. 3
15 5
28
0B 9

25 Therefore, n —
24 24
24. The radii extending to the corners of the
triangle split the circle into three equal parts,
so the measure of angle ADB is
28 7
360 3 — 1200. In radians, this is
27t
24 —I— 28 25 7 28 112 1200 x
1800 3

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25. l@Because triangleABC is45 45 - 90, 28. @Let AD x. ADE is a

Because

AB = 2Vä. Because triangle ABD is 30 — 60 — 90 triangle, DE = xv/j and

2Vä AE 2x. Note that AADE, ABEF, and

30 - 60 - 90, AD and DB is twice ADCF are all congruent.
that: B
DB

We can rationalize the fraction by multiplying

both the top and bottom by US:

DB

26. B Because the triangles are 45 — 45 — 90, c

2
BC The radius of the circle is half BC:
The side length of outer triangle ABC is 3x.
1
The side length of inner triangle DEF is xØ.
Finally, the area of the Because the two triangles are similar, the ratio
of their areas is equal to the square of the
circle is
ratio of their sides:
2

Area ADEF
of (xv6)2 3x2 1

Area of AABC (3x)2 9x2 3

27. To satisfy the triangle inequality theorem,
the third sidemust be less than 5 + 11 — 16
and greater than 11 — 5 6. So the minimum
29. @To satisfy the triangle inequality theorem
(any two sides of a triangle must sum up to be
perimeter is 5 + 11 + 7 — 23 and the
greater than the third side), a must be greater
maximum perimeter is 5 + 11 + 15 — 31. II
than 7 — 5 = 2 and less than 5 7 — 12. b
and Ill are the only values in this range.
must also be greater than 2 and less than 12.

2 < a < 12

2 < b < 12

la — bl cannot be 10, because the difference

between a and b can never be that large, no

matter what values you pick.

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CHAPTER 28 ANSWERS TO THE EXERCISES

30. Because the equilateral triangle lies on a 32. @Draw the height from A as shown below.
side of the square, all their sides are equal, AADB turns out to be a 30 60 90 triangle.
which means A ABE and ADCE are isosceles.
c

600 1200
c
6

The area is

Z AED — 600, ZBAE — ZCDE 300, which 33.

means
Z ABE ZAEB ZDCE — ZDEC 750.

Finally,
ZBEC — 3600 750 750 600 — 1500.

31. D Draw a straight line down the middle.

The length of this line is 9 because the top
part is simply a radius of the semicircle, 0
whose length is half the side of the square, x

-1)

Draw the extra line shown above to form a

30 60 — 90 triangle (the sides are in a ratio
6
of I : 2). The acute angle the line
:

segment forms with the x-axis is 300 which ,

makes 0 360 — 30 3300. In radians, this is

1 ITC
3300 x
1800 6
Using the pythagorean theorem,
34. Because DBCE is a square, DB 3 and
92 + 32 ¯ A 132 triangles ABD and DEO are similar (their
angles are the same). Using the pythagorean
90 AB2
theorem, DO 5. Using the similarity,
90
AD DO
3 10 DB OE
AD 5
3 4
15
AD — 3.75
4

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 THE COLLEGE PANDA

35. @
x
4

12
3x

Triangles ADE and F BE are similar. The sides

of triangle ADE are 3 times longer than the
respective sides of triangle F BE. Because
triangle ABD is a 45 — 45 90 triangle, the
length of BD is 12Vi. If we let BE — x, then
DE 3x.

12v6

x = 3v6

36. @Triangle ABC is a 5 12 13 triangle

(BC 5). Triangle ABC is similar to triangle
AED (the angles are equal). Using this
similarity is tricky because the two triangles
have different orientations. The following is
one example of a correct setup:

DE BC
12
3 5
36
— 7.2
5

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 21: Circles

CHAPTER EXERCISE:

1. @The circumference of the circle is 27tr. The 6. @Because Z BAC is formed from the
square divides the circle into four equal arcs. endpoints of a diameter, its measure is 900.
27Tr nr Since AB — 1 and AC 2, A ABC is a
Therefore, the length of arc A PD is 30 60 90 triangle and Z BAC — 600.
4 2

2. @Finding the radius of each of the small

circles,

2
— 367T
97T

The circumference of the circle is

The radius of the outer circle is equivalent to
27tr = 27T(6) — 1271. Because the equilateral
three radii of the smaller circles, 3 x 3 9.
triangle splits the circumference of the circle
The area is then 8171.
into 3 equal pieces, arc AB is one-third of the
3. @First, find the radius.
circumference: — >< 127T — 471.
2
367T
8. C The area of the circle is
367T. The shaded sector is

107T 5
The circumference of the circle is which means
367T ¯ 18 of the entire circle,
27tr = 27T(6) — 1271. The perimeter of one
5
region is made up of two radii and one-eighth central angle ACB must be of 360.
18
of the circumference.

5
x 3600 1000
6+6+ —(127T) 12 + 1.571 18

Converting this to radians,

57T
1000 x
2 — 497T 180 9

49 We could've gotten this answer directly by

7 sticking to radians. The area of a sector is

1
—r2Ø when 9, the measure of the central
The standard form of a circle with center 2
(h, k) and radius r is (x — + (y — 2
angle, is in radians.
So the equation of the circle is
(x 4-2) 2 -I— Y2 49 1
—r2e 107T
2
5. @The arc measure of AB is twice the 1
measure of the inscribed angle. Therefore, — (6) 29 107T
2
600 1
AB 600, which is — of the 180 107T
3600 6
circumference.

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 THE COLLEGE PANDA

9. 4,5,6, or 7 The arc length can be 12. @Circle P and circle 11 each have an area of
etermine by ro when 0, the measure of the 97T. To get the shaded region, we
central angle, is expressed in radians. need to subtract out the unshaded portions of
Therefore, the arc length must be greater than both circles. Because APHII is equilateral,
3.92 and less than 5 7.85.
ZHPLI and ZHU P are both 600 , which means
the unshaded sectors are each one-sixth of
their respective circles (600 is one-sixth of
We could've done this question by converting
radians back to degrees but the process 3600).

would've taken a lot longer.

1
97T + 97T _ _ (970 157T
10. Draw a square connecting the centers of 6
each circle:

13. @Let y be the angle at the top of the triangle.

2 —
360
7tr 247T

247T
360

36 — 24
10
To get the shaded region, we need to subtract
out the four quarter-circles from the square.
12
The square has an area of 8 x8 = 64. The 10
four quarter-circles make up one circle with 120 —Y
an area of 167T. The area of the
shaded region is then 64 167T. If y is 120, then x and x have to add up to 60.
Therefore, x 30.
11. @Unraveling the cylinder gives a rectangle 14. @From the information given, AB 8,
with a base equal to the circumference and a
BC 4, and because AC is tangent to circle B,
height equal to the height of the cylinder:
ZACB is a right angle. Using the
pythagorean theorem to find AC,

—b 42 — 82

AC2 _ 48

27tr

The surface area of the cylinder is equal to the

The area of
area of this rectangle plus the areas of the two
circles at either end. AABC

27Trh + 27Tr2 27T(4) (5) + 15. @The circle has center (—2, —4) and radius
— 407T + 327T 2. If you draw this circle out, you'll see that
727T
it's tangent only to the y-axis.

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 22: Trigonometry

CHAPTER EXERCISE:

1' Since cosx sin(90 x), 5. @After drawing the right triangle, we let
cos 400 sin 500 a. the opposite side be m and the adjacent side
3 be 1.

2• @Since tanx 0.75 we can draw a

4'
right triangle such that the opposite side is 3
and the adjacent side is 4. m

1
3
Using the pythagorean theorem, the
hypotenuse is m2 + 1. Therefore,
4 m
sin x —
Tti2 I
Using the pythagorean theorem, the
hypotenuse is 5 (this is a 3 — 4 — 5 triangle). 6. @The fact that AB — 5 is irrelevant since
4 the ratios of the sides will always be the same
Therefore, cos x — — —08
5 for proportional triangles. Instead of actually
trying to figure out the lengths of the sides,
3. Since sin 0 cos(90 0) and
let's use a triangle that's easier to work with.
cos0 sin(90 — 0),

sin 0 + cos(90 — 0) 4- cos0 -b sin(90 0) —

sin 9 + sin 0 + cos 0 + cos 0

2 sin 0 + 2 cos 0
3

4. 25 Drawing the triangle,

4 c

Using the pythagoæan theorem, BC 5 (it's

a 3 — 4 — 5 triangle).
30
7
sin B + cos B
5 5 5

c
7. 12

5 1
cos A sin x —
6 4
AC 5 3 1
30 6 4
AC — 25

Cross multiply to get BC 12.

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 THE COLLEGE PANDA

8.
5
Since tan B 2.4
12
, we can let
12. @This question is basically asking for the
13 5 quadrants in which sin 0 can equal cos 9. For
AC — 12 and AB 5. Using the pythagorean them to be equal, they must have the same
theorem, BC — 13. Since the two triangles are sign. That rules out option II since sine is
positive in the second quadrant while cosine
similar,
is negative. In quadrant I, sine and cosine are

AB 5 both positive, and sine is equal to cosine

cos N cos B
when 0 450 (remember your 45 — 45 — 90
BC 13
triangle?). In quadrant Ill, sine and cosine are
both positive, and sine is equal to cosine
sin(90 — x), when 9 2250 (this is the third quadrant
cos 320 sin580. Setting up an equation, equivalent of 450 in the first quadrant).

sin 58 sin 5m — 12

58 —5m 12

70 5m
m¯ 14

10. From the coordinates, AB 5 (—3) —8

and BC 12 — (-3) — 15. Using the
pythagorean theorem to find AC,

AC2 — AB2 + BC2

AC2 82 152

AC2 — 289
17

BC 15
Finally, cos C AC ¯ 17 •

11. D Z ABC measures 900 because it's

inscribed in a semicircle. Therefore, triangle
ABC is a right triangle.
Let the height be AB
and the base be BC. Since the hypotenuse
AC 1,

sin 9

cos 9 BC

1
Area of triangle
2
1
(cos 9) (sin 0)

sin 9 cos 9
2

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 23: Reading Data

CHAPTER EXERCISE:

1. estimate the total commute time for each point:

Point Commute Time

25 + 60 85
B 38 + 40 78
c 45 + 80 — 125
D 80 + 20 — 100

Even though the times were estimated, it's clear that C represents the greatest commute time.

2. C The vertical distance between the points at 2004 and 2006 is the smallest among the answer choices.

3. @The points corresponding to July through September are the highest in both 2013 and 2014.

150 3
250 ¯ 5

5. A San Diego is the only city for which the estimated bar is lower than (to the left of) the actual bar.

6. @Both line graphs go downward every year.

7. @The lowest point with respect to the y-axis is at a little under 40 years of age.
8. @The graph's minimum, 16, must be the weight of the truck when empty. The graph's maximum,
30, must be the weight of the truck at maximum capacity. Subtract the two to get the truck's maximum
capacity, 30 16 14.

9. @From 2010 to 2011, the percent decrease was

30 — 40 1

4
—25% (percent decreases are negative)
40

From 2013 to 2014, the percent increase was

25 - 20 1
— 25%
20 4

2 120 2
10.
3 180 3

11. @Cons01e A generated 250, OOO x 100 — $25, 000, 000. Comsole B generated 225, OOO x 150 — $33, 750, 000.
Console D generated 125, OOO x 250 — $31, 250,000. Console E generated 50, OOO x 300 — $15,000, 000.
Console B generated the most revenue.

12. 3, Company Y's profit was about 6 million and Company X's profit was about 12 million
Quarter
(twice Company Y's). In no other quarter was Company X's profit as close to being twice Company Y's.

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 THE COLLEGE PANDA

13. c Alabama spent a combined 15 + 2.5 17.5 billion. Alaska spent 7.5 + 7.5 15 billion. Arizona spent
12.5 + 7.5 20 billion. Arkansas spent 10 +5 — 15 billion. Arizona spent the most.

14. 44 During the first two hours, Jeremy answered 4 calls per hour for a total of 2 x 4 —
8 calls. During
the next three hours, Jeremy answered 8 calls per hour for a total of 3 x 8 = 24 During the final
calls.

two hours, Jeremy answered 6 calls per hour for a total of 2 x 6 12 calls. He answered a total of
8 + 24 12 — 44 calls.
15. From the graph, we can see that it takes Greg's glucose levels 2.5 hours to return to their initial value
(140 mg/dL) after breakfast and 8 — 4 = 4 hours to return to their initial value (also 140 mg/dL) after
dinner.
4 2.5 = 1.5

16. 6 At 30 miles per hour, Car X gets 25 miles per gallon. Driving for 5 hours at 30 miles per hour covers
a total distance of 5 x 30 — 150 miles.

1 gallon
150 miles x 6 gallons
25 miles

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 24: Probability

CHAPTER EXERCISE:

Stop sign violations committed by truck drivers 39

0.433
Stop sign violations 90

2. D
Plumbers with at least 4 years of experience 40, 083 + 45, 376
46
All plumbers 183, 885

Plumbers with at least 4 years of experience 40,083 45,376

0.22
Workers with at least 4 years of experience 182, 410 + 208, 757

4. D The percentage of silver cars is 100 — 20 —33 10 14 23. Red and silver make up 20 23 _ 43
percent of the cars.

Games won as underdogs 10 2

Games played as underdogs 45 9

6. @Filling in the table,

Week 1 Week 2 Week 3 Total

Box springs 35 40 20 55 150

Mattresses 47 61 68 22 198
Total 82 101 88 77 348

Box spring units sold during weeks 2 and 3 40 + 20 2

All box spring units sold 150 5

29 32
7. @For the USA, the probability is 104
0.28. For Russia, the probability is
82
0.39. For Great Britain,

19 14
the probability is —
65
0.29. For Germany, the probability is —
44
0.32. The country with the highest

probability is Russia.

Cartilaginous fish species in the Philippines Cartilaginous fish species in New Caledonia
Total fish species in the Philippines Total fish species in New Caledonia
400 300 1 1 2
400+ 800 300 + 1,200 5 15

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 THE COLLEGE PANDA

9. Filling in the table,

Lightning-caused fires Human-caused fires Total

East Africa 55 65 120

South Africa 30 70 100

Total 85 135 220

Human-caused fires in East Africa 65 13

Fires in East Africa 120 24

10.

Defective from Assembly Line A 300 3

Defective 800 8

II. @
Duplex with 2 family members or less 22 + 12 17

Duplex 23

12. @The total number of samples contaminated with Chemical A is (450 x 0.08) + (550 x 0.06) — 69.
Contaminated samples 69
— 0.069
All samples 1, ooo

13. @The test is incorrect when it gives positive indicators for patients who don't have the virus and
negative indicators for patients who do, a total of 30 + 50 — 80 occurrences.

80 8
1000 ¯ 100 80/0

14. @The number of patients cured by the sugar pill is 90 3 30. The number of patients who weren't
5
cured by the sugar pill is 30 x — 75.
2

Cured Not cured

Drug 90 25

Sugar Pill 30 75

Given a sugar pill and cured 30 2

Given a sugar pill 30 + 75 7

15. 240 Let the number of seniors who prefer gym equipment be x.

x 1

Cross multiplying,

X + 160
2x 160
x — 80

There are 80 + 160 240 seniors at the school.

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 25: Statistics I

CHAPTER EXERCISE:

1. The sum of the heights in the first class is 14 x 63 882. The sum of the heights in the second class is

21 x 68 — 1, 428. The sum of the heights in the combined class is then 882 -b 1428 2, 310. The average
height is

Sum of the heights 2, 310

66
Total number of students 14 -b 21

2. @The sum of all five of Kristie's test scores is 5 x 94 470. The sum of her last three test scores is

3 x 92 = 276. The difference between these two sums is the sum of her first two test scores: 470 — 276
194
194. The average of her first two test scores is then — 97.
2

3. @Because there are 20 editors, the median is the average of the 10th and 11th editors' number of books
read. From the graph, the 10th and 11th editors both read 10 to 15 books last year, which means the
average must also be between 10 and 15. The only answer choice between 10 and 15 is 12.

(18 x 6) + (19 x 3) + (20 x 5) + (21 x 4) + (22 x 2) + (23 x 3) + (24 x 1)

— 20.25
24 24

5. @The standard deviation decreases the most when the outliers, the data points furthest away from the
mean, are removed. The outliers here are the Rhone and the Vosges.

6. Even though the frequencies are the same, the travel times themselves are more spread out for Bus
B. The travel times for Bus A are much closer together. Therefore, the standard deviation of travel times
for Bus A is smaller.

7. C The median weight is represented by the 10th kayak (47 pounds for both Company A and Company
B). The median weight is the same for both companies.

8. The median is represented by the average of the 14th and 15th days, both of which are 670 F.

9. @By definition, at least half the values are greater than or equal to the median and at least half the
values are less than or equal to the median.

10. range of 3 days means the difference between the longest shelf life and the shortest shelf life among
the units is 3. This could be 10 days vs. 13 days or 25 days vs. 28 days. The range says nothing about the
mean or median.

(5 x 2) (6 x I) (8 x 4) —b (9 x 2) —b (10 x I) 76
Mean — 7.6
2+1 +4+2+1 10

Range — 10 — 5 5

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 THE COLLEGE PANDA

12. B Arranging the scores in order,

75, 83, 87, 87, 90, 91, 98

75 83 87 + 87 + 90 + 91 4_ 98
The average is 87.3. The mode is 87. The median is also 87. The
7
range is 98 75 23. From these numbers, I is false, II is false, and Ill is true.

13. @Based on the heights of the bars in the graph, the mean in School A is clearly higher. The median in
School A is represented by the 10th class (4 films) and the median in School B is represented by the 8th
class (also 4 films). The median is the same for both schools.

14. @Before the 900-calorie meal is added, the median is the average of the 5th and 6th meals (550), the
mode is 550, and the range is 900 — 500 400. After the 900-calorie meal is added, the median becomes
the 6th meal (still 550), the mode is still 550, and the range is still 400. None of them change.

15. @Before the car is removed, the median is represented by the 8th car (23 mpg). After the car is removed,
themedian is represented by the average of the 7th and 8th cars (still 23 mpg). So the median stays the
same. However, the mean and the standard deviation both decrease. We're removing a data point higher
than all the others so the mean decreases. We're also reducing the spread in the data so the standard
deviation decreases.

16. First, it's easy to see that the mean will decrease since we're replacing the maximum data point with
a minimum. Now before the replacement, the range is 90 — 45 45. After the replacement, the range
is 65 — 20 45, so the range remains the same. Before the replacement, the median is represented by
the average of the 9th and 10th cars (57). After the replacement, the median is represented by 10th car
(still 57, don't forget to count the replacement as the first value). The median also remains the same.
Therefore, the mean changes the most.

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 26: Statistics II

CHAPTER EXERCISE:

1. @There are 2 points above the line of best fit when the value along the x-axis is 19.

2. C The line of best fit gives a y-value of 55 when the x-value is 75.

3. C First, the survey should be conducted with students from the university's freshman class since that's
the intended target. Secondly, the larger the sample, the more valid the results.
110
4. Using proportions, Candidate A is expected to receive 250
x 500, OOO 220, OOO votes. Candidate B

is expected to receive x 500, 000 = 280, 000 votes. So Candidate B is expected to receive 280, 000
250
220, OOO 60, OOO more votes.
5. The y-intercept is the value of y when the value of x is 0. In this case, it's the average shopping time
when the store discount is 00/0 (no discount).

6. @The slope is rise over run. Because the line of best fit has a positive slope, it's the increase in revenue
for every dollar increase in advertising expenses.Note that because both revenue and advertising expenses
are expressed in thousands of dollars in the graph, they cancel out and have no effect on the interpretation
of the slope. That's why the answer isn't (B).

7. B The slope is rise over run. Because the line of best fit has a positive slope, it's the increase in box office
sales per minute increase in movie length.

8. @The y-intercept is the value of y when the value of x is 0. In this case, it's the expected number of
mistakes made when the cash prize is 0 dollars (no cash prize).

9. @This question is asking for the slope of the line of best At 20 grams of fat, there are 340 calories. At
fit.

25 grams of fat, there are 380 calories. Calculating the slope from two points,

380 -340 40
8
25 20 5

10. @The oat field whose yield is best predicted by the line of best fit is represented by the point closest to
the line. That point has an x-value of 350, which is the amount of nitrogen applied to that field.

11. @The point farthest from the line of best fit is at an x-value of 7. The total number of seats at the food
court represented by this point is 7 x 80 560.

12. D To draw a reliable conclusion about the effectiveness of the new vaccine, the patients must be
randomly assigned to their treatment. Only answer (D) leads to random assignment. Note that answer
(C) does not because the patients are allowed to group themselves as they desire. For example, three
friends might want to remain in the same group, leading to assignment that is not random.

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 THE COLLEGE PANDA

13. @Answer (A) is wrong because it's Week 1 had an

possible that most of the basketballs produced in
air pressure of over 8.2 psi. Week 2. We don't know for sure. Answer (B) is wrong because
Likewise for
it's too definite. Just because the sample means were 0.5 psi apart doesn't mean the true means, which

would take into account all the basketballs produced in Week 1 and Week 2, were also 0.5 psi apart.
That's why there's a margin of error for the samples. Answer (D) is wrong because the samples suggest
the reverse: the mean air pressure for Week 1 (8.2 psi) is greater than the mean air pressure for Week 2
(7.7 psi). Answer (C) is correct because the greater the sample size, the lower the margin of error. The
sample from Week 1 had a lower margin of error than the sample from Week 2.

14. D The lower the standard deviation (variability), the lower the margin of error. Selecting students who
are following the same daily diet plan will likely lead to the lowest standard deviation because they are
likely to be eating the same number of servings of vegetables. The other answer choices would result in
much more variability.

15. C Answer (C) best expresses the meaning of a confidence interval, which applies only to the statistical
mean and does not say anything about blue—spotted salamanders themselves. Answer (D) is wrong
because the study involved only blue-spotted salamanders, not all salamanders.

16. B The most that we can conclude is that there is a negative association between the price of food and
the population density in U.S. cities (as one goes up, the other goes down). We CANNOT conclude that
there is a cause and effect relationship between the two. We can't say that one causes the other.

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CHAPTER 28 ANSWERS TO THE EXERCISES

Chapter 27: Volume

CHAPTER EXERCISE:

1. @Each piece is half the cylinder. 5. This question is asking for the volume of

the cylinder. The radius of the base is 2 and

1 1 since the diameter of each tennis ball is 4, the
—V — —nr2h 107T
2 2 height of the cylinder is 4 x 3 12.

V— 7tr2h — 487T
2. B The height of the box is 100 : 25=4
(dividing the volume by the area of the base
gives us to the height). The sides of the base 6. The shortest way to do this question is to
are 25 5 inches long. The rectangular box pretend that the block is liquefied and poured
has dimensions 5 x 5 x 4. into the aquarium. How high would the level
of the liquid rise?

4 V- lwh

5,000 — (80) (25)h

5
2.5 h
5
The longer way do this question is to find
to
The top and bottom have a surface area of the original volume, add the block, find the
2(5 x 5) 50. The front and back have a
new height, and then compare it to the
surface area of 2(5 x 4) — 40. The left and
original height. While not the fastest method,
right have a surface area of 2(5 x 4) 40.
it is certainly viable.
The total surface area is 50 + 40 + 40 130.

3. @Let the side of the cube be s. The cube has

7. A If you take away all the cubes with black
six faces and the area of each face is s paint on them, you are essentially uncovering

Solving for s in terms of a,

an inner cube with a side length of 3. front A
view is shown below.
6s 2 — 24a 2
2 2
s

The volume is then s3 — (2a)3 8a3.

There are 33 27 cubes that are unpainted.
4. The cylindrical tank that can be filled in 3
hours has a volume of = 967T. The
tank in question has a volume of
(8) = 2887T. Using the first tank as a
conversion factor,

3 hours
2887T X 9 hours
967t

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 THE COLLEGE PANDA

8. Since each small cube has a volume of 10. @Draw a line down the middle of the cone
23 and the volume of the outer box is
8 to form a right triangle with the radius and
83 512, there must be 512 + 8 — 64 cubes in the slant height. This triangle is a multiple of
the box. If you take away all the cubes that the 3 — 4 — 5 right triangle: 9 — 12 15. You

are touching the box, you are essentially could've used the pythagorean theorem
uncovering an inner rectangular box with a instead if you weren't aware of this. In any
square base of side 4 and a height of 6. A case, the height of the cone is 12 cm.
front view is shown below.
V volume of cone + volume of hemisphere

V — 7tr2h +
8 nr3)

V
4

8 V — 3247T + 4867T 8107T

8

The volume of this inner rectangular box is 11. @This question is essentially asking for the
4 x4 x 6 96. Since each cube has a volume volume, or the amount of room, in the crate.
of 8, there are 96 : 8 — 12 cubes that are not The room in the crate can be seen as a
touching the box, which means there are rectangular box with a length of
64 — 12 = 52 cubes that are touching. You 10 —1—1 8 inches, a width of 8 — 1 — 1 6
also could've taken the straight-forward inches, and a height of 3 1 = 2 inches.
approach of counting up the cubes along the
sides. If you took this route, you should've v 96
gotten something along the lines of
16 16 + 8+4 — 52. 12. @Cut the staircase vertically into 3 blocks.
9. @The only cubes that have exactly one face
painted black are the ones in the middle of
Volume of staircase — Volume of block 1

each side. For example, the front side has + Volume of block 2
3 x 1 3 of these cubes. Volume of block 3

3 V- (5 x2 x 0.2) + (5 x 2 x 0.4) + (5 x 2 x 0.6)

3
12
4
5
Mass Density x Volume — 130 x 12 1, 560 kg
The right side has 2 x 1 2 of these cubes,
and the top has 3 x 2 — 6 of these cubes. So
far, we have 3 + 2 + 6 — 11 of these cubes. To

account for the back, left, and bottom sides,

we double this to get 22 cubes.

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