BASIC ALGEBRA

1. If Lynn can type a page in p minutes, what piece of the page can she do in 5 minutes?

1. 5/p

2. p – 5

3. p + 5

4. p/5

5. 1- p + 5

2. If Sally can paint a house in 4 hours, and John can paint the same house in 6 hour, how long will it
take for both of them to paint the house together?

1. 2 hours and 24 minutes

2. 3 hours and 12 minutes

3. 3 hours and 44 minutes

4. 4 hours and 10 minutes

5. 4 hours and 33 minutes

3. Employees of a discount appliance store receive an additional 20% off of the lowest price on an
item. If an employee purchases a dishwasher during a 15% off sale, how much will he pay if the
dishwasher originally cost $450?

1. $280.90

2. $287.00

3. $292.50

4. $306.00

5. $333.89

4. The sales price of a car is $12,590, which is 20% off the original price. What is the original price?

1. $14,310.40

2. $14,990.90

3. $15,290.70

4. $15,737.50

5. $16,935.80

5. Solve the following equation for A : 2A/3 = 8 + 4A

1. -2.4
 2. 2.4

3. 1.3

4. -1.3

5. 0

6. If Leah is 6 years older than Sue, and John is 5 years older than Leah, and the total of their ages is
41. Then how old is Sue?

1. 8

2. 10

3. 14

4. 19

5. 21

7. Alfred wants to invest $4,000 at 6% simple interest rate for 5 years. How much interest will he
receive?

1. $240

2. $480

3. $720

4. $960

5. $1,200

8. Jim is able to sell a hand-carved statue for $670 which was a 35% profit over his cost. How much did
the statue originally cost him?

1. $496.30

2. $512.40

3. $555.40

4. $574.90

5. $588.20

9. The city council has decided to add a 0.3% tax on motel and hotel rooms. If a traveler spends the
night in a motel room that costs $55 before taxes, how much will the city receive in taxes from him?

1. 10

2. 11 cents

3. 15 cents
 4. 17 cents

5. 21 cents

10. A student receives his grade report from a local community college, but the GPA is smudged. He
took the following classes: a 2 hour credit art, a 3 hour credit history, a 4 hour credit science course, a
3 hour credit mathematics course, and a 1 hour science lab. He received a “B” in the art class, an “A”
in the history class, a “C” in the science class, a “B” in the mathematics class, and an “A” in the science
lab. What was his GPA if the letter grades are based on a 4 point scale? (A=4, B=3, C=2, D=1, F=0)

1. 2.7

2. 2.8

3. 3.0

4. 3.1

5. 3.2

11. Simon arrived at work at 8:15 A.M. and left work at 10: 30 P.M. If Simon gets paid by the hour at a
rate of $10 and time and ½ for any hours worked over 8 in a day. How much did Simon get paid?

1. $120.25

2. $160.75

3. $173.75

4. $180.00

5. $182.50

12. Grace has 16 jellybeans in her pocket. She has 8 red ones, 4 green ones, and 4 blue ones. What is
the minimum number of jellybeans she must take out of her pocket to ensure that she has one of
each color?

1. 4

2. 8

3. 12

4. 13

5. 16

13. If r = 5 z then 15 z = 3 y, then r =

1. y

2. 2y

3. 4y
 4. 10y

5. 15y

14. If 300 jellybeans cost you x dollars. How many jellybeans can you purchase for 50 cents at the
same rate?

1. 150/x

2. 150x

3. 6x

4. 1500/x

5. 600x

15. Lee worked 22 hours this week and made $132. If she works 15 hours next week at the same pay
rate, how much will she make?

1. $57

2. $90

3. $104

4. $112

5. $122

16. If 8x + 5x + 2x + 4x = 114, the 5x + 3 =

1. 12

2. 25

3. 33

4. 47

5. 86

17. You need to purchase a textbook for nursing school. The book cost $80.00, and the sales tax where
you are purchasing the book is 8.25%. You have $100. How much change will you receive back?

1. $5.20

2. $7.35

3. $13.40

4. $19.95

5. $21.25
18. You purchase a car making a down payment of $3,000 and 6 monthly payments of $225. How
much have you paid so far for the car?

1. $3225

2. $4350

3. $5375

4. $6550

5. $6398

19. Your supervisor instructs you to purchase 240 pens and 6 staplers for the nurse’s station. Pens are
purchased in sets of 6 for $2.35 per pack. Staplers are sold in sets of 2 for 12.95. How much will
purchasing these products cost?

1. $132.85

2. $145.75

3. $162.90

4. $225.25

5. $226.75

20. If y = 3, then y3(y3-y)=

1. 300

2. 459

3. 648

4. 999

5. 1099
Answers & Explanations

1. A

The following proportion may be written: 1/p=x/5. Solving for the variable, x, gives xp = 5, where x=5/p.
So, Lynn can type 5/p pages, in 5 minutes.

2. A

Sally can paint 1/4 of the house in 1 hour. John can paint 1/6 of the same house in 1 hour. In order to
determine how long it will take them to paint the house, when working together, the following equation
may be written: 1/4 x+1/6 x=1. Solving for x gives 5/12 x=1, where x= 2.4 hours, or 2 hours, 24 minutes.

3. D

Sale Price = $450 – 0.15($450) = $382.50, Employee Price = $382.50 – 0.2($382.50) = $306

4. D

$12,590 = Original Price – 0.2(Original Price) = 0.8(Original Price), Original Price = $12,590/0.8 =
$15,737.50

5. A

In order to solve for A, both sides of the equation may first be multiplied by 3. This is written as
3(2A/3)=3(8+4A) or 2A=24+12A. Subtraction of 12A from both sides of the equation gives -10A=24.
Division by -10 gives A = -2.4.

6. A

Three equations may initially be written to represent the given information. Since the sum of the three
ages is 41, we may write, l + s + j = 41, where l represents Leah’s age, s represents Sue’s age, and j
represents John’s age. We also know that Leah is 6 years older than Sue, so we may write the equation, l
= s + 6. Since John is 5 years older than Leah, we may also write the equation, j = l + 5. The expression for
l, or s + 6, may be substituted into the equation, j = l + 5, giving j = s + 6 + 5, or j = s + 11. Now, the
expressions for l and j may be substituted into the equation, representing the sum of their ages. Doing
so gives: s + 6 + s + s + 11 = 41, or 3s = 24, where s = 8. Thus, Sue is 8 years old.

7. E

Simple interest is represented by the formula, I = Prt, where P represents the principal amount, r
represents the interest rate, and t represents the time. Substituting $4,000 for P, 0.06 for r, and 5 for t
gives I = (4000)(0.06)(5), or I = 1,200. So, he will receive $1,200 in interest.

8. A

$670 = Cost + 0.35(Cost) = 1.35(Cost), Cost = $670/1.35 = $496.30

9. D
The amount of taxes is equal to $55*0.003, or $0.165. Rounding to the nearest cent gives 17 cents.

10. C

The GPA may be calculated by writing the expression, ((3*2)+(4*3)+(2*4)+(3*3)+(4*1))/13, which equals
3, or 3.0.

11. C

From 8:15 A.M. to 4:15 P.M., he gets paid $10 per hour, with the total amount paid represented by the
equation, $10*8=$80. From 4:15 P.M. to 10:30 P.M., he gets paid $15 per hour, with the total amount
paid represented by the equation, $15*6.25=$93.75. The sum of $80 and $93.75 is $173.75, so he was
paid $173.75 for 14.25 hours of work.

12. D

If she removes 13 jellybeans from her pocket, she will have 3 jellybeans left, with each color
represented. If she removes only 12 jellybeans, green or blue may not be represented.

13. A

The value of z may be determined by dividing both sides of the equation, r=5z, by 5. Doing so gives
r/5=z. Substituting r/5 for the variable, z, in the equation, 15z=3y, gives 15(r/5)=3y. Solving for y gives r =
y.

14. A

50 cents is half of one dollar, thus the ratio is written as half of 300, or 150, to x. The equation
representing this situation is 300/x*1/2=150/x.

15. B

The following proportion may be used to determine how much Lee will make next week: 22/132=15/x.
Solving for x gives x = 90. Thus, she will make $90 next week, if she works 15 hours.

16. C

The given equation should be solved for x. Doing so gives x = 6. Substituting the x-value of 6 into the
expression, 5x + 3, gives 5(6) + 3, or 33.

17. C

The amount you will pay for the book may be represented by the expression, 80+(80*0.0825). Thus, you
will pay $86.60 for the book. The change you will receive is equal to the difference of $100 and $86.60,
or $13.40.

18. B

The amount you have paid for the car may be written as $3,000 + 6($225), which equals $4,350.

19. A
You will need 40 packs of pens and 3 sets of staplers. Thus, the total cost may be represented by the
expression, 40(2.35) + 3(12.95). The total cost is $132.85.

20. C

Substituting 3 for y gives 33 (33-3), which equals 27(27 – 3), or 27(24). Thus, the expression equals 648.
Intermediate Algebra

1. Simplify the expression (4x + 22x) / (2x)

A6
B 2+2x
C 2×2x
D 2x+1

2. Simplify the expression (2x2-5x-12)/(2x2-4x-16).

A (x-6)/2(x-2)
B (x-6)/2(x+2)
C (2x+3)/2(x-2)
D (2x+3)/2(x+2)

3. Suppose that the function f(x) is a quadratic function with roots at x=2-3i and x=2+3i. Find f(x).

A f(x)=x2-4x-5
B f(x)=x2-4x+13
C f(x)=x2-6ix-5
D f(x)=x2-6ix+13

4. Solve the inequality for x. Select all that apply.

4x3+10x2-24x<0

A x<-4
B -4
C0
D x>3/2

5. A baseball is thrown up in the air from an initial height of 6 feet. Its height above the ground (in
feet) t seconds after being thrown is given by the function h(t)=-16t2+46t+6. How long will it take (in
seconds) for the baseball to hit the ground?

A 2 seconds
B 5/2 seconds
C 3 seconds
D 4 seconds

6. Solve the equation for x. Select all that apply.

log2(8x-x2 )=4

A x=-8
B x=0
C x=4
D x=8

7. Calculate the average rate of change of f between x=1 and x=4.

f(x)=x3+3x+1
A6
B 20/3
C 24
D 72

8. Simplify the expression (x3-3x2+2x-6)/(x2-9).

A1
B (x-3)/(x+3)
C (x2+2)/(x-3)
D (x2+2)/(x+3)

9. Suppose that angle ? is in Quadrant I and cos ? = 12/13. Find tan ?.

A tan ? = 1/13
B tan ? = 13
C tan ? = 5/12
D tan ? = 12/5

10. Which expression is equivalent to 6√x+10x?

A 2(3x-1+5x)
B 2(3x1/2+5x)
C 2x(3x-1+5)
D 2x(3x1/2+5)

Answers & Explanations

1. D

You can solve this problem either (1) by simplifying the numerator and denominator separately and then
simplifying the result or (2) by using the distributive property. For this problem, we will use the first
method.
First rewrite 4x as an exponent of 2 using the property, (bx )y=bxy.

4x=(22 )x=22x
Then use this to simplify the numerator with the property, bx x y = bx+y.

Finally, simplify the result using bx / by = bx-y

2. D

To simplify the expression, first factor the numerator and the denominator. By the trial-and-error
method, the numerator can be factored into two binomials as follows.

2x2 – 5x – 12 = (2x + 3)(x -4 )

For the denominator, factor out the common factor, which is 2.

2x2 – 4x – 16
=2(x2 – 2x -8)
=2(x-4)(x+2)

Thus, the factored form of the expression is

Notice that there is a common factor, (x – 4), which is in both the numerator and the denominator.
Therefore, you can further simplify the expression by cancelling it out.
3. B

The roots of a quadratic function f(x) are the values of x for which f(x)=0. A quadratic function written in
the form f(x)=(x-a)(x-b) has roots at x=a and x=b. Therefore, to find f(x), substitute 2-3i and 2+3i for a
and b into this equation and simplify the result. Note that (2-3i)(2+3i)=4-9i2=13.

f(x)=(x-a)(x-b)
=[x-(2-3i)][x-(2+3i)]
=x2-(2-3i)x-(2+3i)x+(2-3i)(2+3i)
=x2-2x+3ix-2x-3ix+13
=x2-4x+13

4. A and C

To solve, first factor the polynomial. Notice that the greatest common factor (GCF) of the terms is 2x.
Factor this expression out and then use trial-and-error to factor the resulting trinomial.

4x3+10x2-24x
=2x(2x2+5x-12)
=2x(2x-3)(x+4)

Solving for 0, we find that the roots of the polynomial are x=0, x=3/2, and x=-4.

These values divide the number line into four intervals. Choose a test number from each interval and
determine whether the product is positive or negative. For this problem, we will use -5, -1, 1, and 2 as
test numbers. Substitute these values into the original polynomial.

x=-5:
4(-5)3+10(-5)2-24(-5)
=-375+250+120
=-5

x=-1:
4(-1)3+10(-1)2-24(-1)
=-4+10+24
=30

x=1:
4(1)3+10(1)2-24(1)
=4+10-24
=-10

x=2:
4(2)3+10(2)2-24(2)
=32+40-48
=24

Thus, the given inequality, 4x3+10x2-24x<0, is satisfied by numbers less than -5 and numbers between 0
and 3/2.

5. D

The baseball will hit the ground when its height is zero. In mathematical notation, this will happen when
h(t)=0. Therefore, we need to set the given function equal to zero.

h(t)=0
-16t2+46t+6=0

Now solve the resulting equation. Factor the left side and use the zero-product property to solve for t.

-2(8t2-23t-3)=0
-1(8t+1)(t-3)=0
t=-1/8 t=3

The answer only makes sense when t is positive, so we can discard the negative value. Thus, the
calculator will hit the ground 3 seconds after it is thrown.

6. C

nly: The logarithm of a number is the exponent that the base must to be raised to in order to get that
number. For example, since 23=8, it is also true that log_2?8=3. Thus, the given equation log_2?(8x-
x2 )=4 implies that

8x-x2=24

Simplify this equation and solve for x.

8x-x2=16
0=x2-8x+16
0=(x-4)2
x=4

Thus, the solution is x=4. Check this value on your own by substituting it into the original equation to
make sure that the result is a true statement.

7. C

The average rate of change of a function f between x=a and x=b can be computed with the formula
Average rate of change=(f(b)-f(a))/(b-a)
To use this formula, first calculate f(1) and f(4).

Then use these values to calculate the average rate of change.

Average rate of change=(f(4)-f(1))/(4-1)

8. D

To simplify the expression, first factor the numerator and the denominator. The numerator can be
factored by grouping as follows.

x3-3x2+2x-6
=x2 (x-3)+2(x-3)
=(x2+2)(x-3)

For the denominator, factor using the difference of squares formula, a2-b2=(a+b)(a-b).

x2-9=(x+3)(x-3)

Thus, the factored form of the expression is

(x3-3x2+2x-6)/(x2-9)
=(x2+2)(x-3)/(x+3)(x-3)

Notice that there is a common factor, (x-3), which is in both the numerator and the denominator.
Therefore, you can further simplify the expression by cancelling it out.

(x2+2)(x-3)/(x+3)(x-3)
=(x2+2)/(x+3)

9. C

Use a unit circle to model the value of cosine. In a right triangle, the cosine function is
cos?=adjacent/hypotenuse. Using the Pythagorean Theorem, we find that the length of the second leg is

b=√(c2-a2 )
=√(12-(12/13)2 )
=√(25/169)
=5/13

Since the tangent function is tan?=opposite/adjacent, the value of tan? is tan?=√(5/13)/(12/13)=√5/12

In Quadrant I, the values of cosine and tangent are both positive. Therefore, tan?=5/12.
10. B

All of the choices involve two transformations of the given expression: factoring out either 2 or 2x and
changing the radical to an exponent. First factor out the greatest common factor (GCF) of the terms. In
this case, the GCF is 2.

6√x+10x
=2(3√x+5x)

In addition, the square root of x is equal to x raised to the 1/2 power.

2(3√x+5x)
=2(3x1/2+5x)
ADVANCED ALGEBRA

1. (2a2b – 3c3)(3a3b + 4c) =

1. 5a6b2 + 12c4 – 9a3bc3 – 12c4

2. 5a5b2 + 8a2bc – 9a3bc3 + 12c4

3. 6a5b2 + 8a2bc – 9a3bc3 + 12c4

4. 6a6b2 + 8a2bc – 9a3bc3 – 12c4

5. 6a5b2 + 8a2bc – 9a3bc3 – 12c4

2. Which equation is represented by the graph shown below?

1. 5a6b2 + 12c4 – 9a3bc3 – 12c4

2. 5a5b2 + 8a2bc – 9a3bc3 + 12c4

3. 6a5b2 + 8a2bc – 9a3bc3 + 12c4

4. 6a6b2 + 8a2bc – 9a3bc3 – 12c4

5. 6a5b2 + 8a2bc – 9a3bc3 – 12c4

3. A function f(x)= 2×2 + 7 is defined by. What is the value of 2f(x)- 3 ?

A. 4x2 + 11

B. 4x4 + 11

C. x2 + 11

D. 4x2 + 14

E. 2x2 + 14
4. A straight line with slope +4 is plotted on a standard Cartesian (xy) coordinate system so that it
intersects the y-axis at a value of y = 1. Which of the following points will the line pass through?

A. (2,9)

B. (0,-1)

C. (0,0)

D. (4,1)

E. (1,4)

5. A package is dropped from an airplane. The height of the package at any time t is described by the
equation, y(t)= -1/2 at2 + h0 where y is the height,h0 is the original height, and a is the acceleration
due to gravity. The value of a is 32ft/sec2. If the airplane is flying at 30,000 feet, what is the altitude of
the package 15 seconds after it is dropped?

A. 26400 ft

B. 22800 ft

C. 15640 ft

D. 7200 ft

E. 0 ft

6. Given the equations for two straight lines, ax + 3y = 18, and 15x + ay = 24,what positive value of the
constant a would make the lines parallel in the standard xycoordinate plane?

A. 6

B. √45

C. √18

D. 452

E. 62

7. Which of the following could be a graph of the function y = 1/x ?

1.

2.
3.

4.
 5.
8. A tire on a car rotates at 500 RPM (revolutions per minute) when the car is traveling at 50 km/hr
(kilometers per hour). What is the circumference of the tire, in meters?
 A. 50,000/2π

B. 50,000/(60*2π)

C. 50,000/(500*2π)

D. 50,000/60

E. 10/6

9. Which of the following expressions is equivalent to (a+b)(a-b) ?

A. a2-b2

B. (a+b)2
 C. (a-b)2

D. ab(a-b)

E. ab(a+b)

Answers and Explanations

1. E

To multiply two binomials, use the FOIL method. FOIL stands for First, Outside, Inside, Last. When
multiplying each pair of terms, remember to multiply the coefficients, then add the exponents of each
separate variable. So, the product of the First terms is 2a2 b*3a3b = 6a5b2. The product of the Outside
terms is 2a2b*4c=8a2 bc. The product of the Inside terms is -3c3*3a3 b=-9a3 bc3 . The product of the
Last terms is-3c3*4c=-12c4. The final answer is simply the sum of these four products.

2. C

The line in the graph has a negative slope and a positive y-axis intercept, so the factor multiplying the
variable x, or the slope, must be negative, and the constant, or y-intercept, must be positive.

3. A

Evaluate as follows: 2f(x)-3=2(2×2+7)-3=4×2+14-3=4×2+11.

4. A

As defined, the line can be described by the equation. Expression A fits this equation: . The others do
not.

B. -1≠4*0+1

C. 0≠4*0+1

D. 1≠4*4+1

E. 4≠4*1+1

5. A

Simply evaluating the expression yields

y(15)=-1/2*32*(15)2+30,000
=-16*225+30,000
=-3600+30,000=26400.

Since this value is unique, all the other answers are incorrect.

6B
Rearranging each equation so that it shows y as a function of the variable x, we have y=-(a/3) x+18/3
and y=-(15/a)x+24/a.

The lines will be parallel if the slopes, or the coefficients multiplied times x, are equal. Therefore -(a/3)=-
(15/a), which yields a2=45, and taking the square root yields a=√45.

Since this value is unique, all the other answers are incorrect.

7. A

This is a typical plot of an inverse variation, in which the product of the dependent and independent
variables, x and y, is always equal to the same value. In this case the product is always equal to 1, so the
plot occupies the first and third quadrants of the coordinate plane. As x increases and approaches
infinity, ydecreases and approaches zero, maintaining the constant product. In contrast, answer B is a
linear plot corresponding to an equation of the form y=x. C is a quadratic plot corresponding to y=x2. D is
an exponential plot corresponding to y=2x. E is another linear plot corresponding to y=x/4+1 .

8. E.

t is not necessary to use the circle circumference formula to solve the problem. Rather, note that 50
km/hr corresponds to 50,000 meters per hour. We are given the car tire’s revolutions per minute and
the answer must be represented as meters; therefore, the speed must be converted to meters per
minute. This corresponds to a speed of 50,000/60 meters per minute, as there are 60 minutes in an
hour. In any given minute, the car travels 50,000/60 meters/min, and each tire rotates 500 times
around, or 500 times its circumference. This corresponds to 50,000/(60*500)=10/6 meters per
revolution, which is the circumference of the tire.

9. A:

Compute the product using the FOIL method, in which the First terms, then the Outer terms, the Inner
terms, and finally the Last terms are figured in sequence of multiplication. As a result (a+b)(a-b)=a2-
ab+ba-b2,. Since ab is equal to ba, the middle terms cancel out each other which leaves a2– b2.