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TABLE OF CONTENTS

PTS Tutorial and Review Systems Katipunan Branch 2014

ALGEBRA OF NUMBERS AND INTEGERS

Real Number System .

Properties of Real Numbers........................................................ .

Basic Operations of Real Numbers and Summary of Key Concepts-.........................................................................5

DECIMAL, FRACTIONS AND
SCIENTIFIC
NOTATION .................................10
RATIO AND
PROPORTION........................................................................................................................................,...II
VARIATIONS 12 STATISTICS
13
Mean , Median And Mode ............................................,............................ .

Permutations, Combinations And Probability.................................................„..

NUMBER SERIES AND PROGRESSIONS18

DATA INTERPRETATION .....................................................21

BASIC MATH TEST

BASIC MATH TEST 31
DATA INTERPRETATION EXERCISES
ALGEBRA40

Polynomials/Algebraic
Special Products and Factoring
Rational Expression
Complex Fraction
Radicals
Linear Equations
Literal Equations
System of linear Equation in 2 Variables...............--.........„....-.......
Quadratic Equation
Inequalities...... .... .. ..........................52
Relations And Functions 55
Exponential And Logarithmic .
Complex Numbers
ALGEBRA TEST A .58
ALGEBRA TEST B .....
PLANE GEOMETRY ..69

Basic terms of geometry .

. ...........69
Kinds of
Understanding Polygons in General.

Mathematics 2014
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09155057703 3998584
 Understanding Circles ........71
Understanding Triangles 72
Understanding Quadrilaterals ............72
PTS Tutorial and Review Systems Katipunan Branch 2014

Area And Volume . .. 77

ANALYTIC 78
Rectangular Coordinate System .
78
Lines and First-Degree Equations...... . ....................78
Conics 80
GEOMETRY TEST A.
GEOMETRY TEST .........................92
101
TRIGONOMETRIC/ CIRCULAR FUNCTIONS ........................ . 101
Pythagorean Theorem .101
Algebraic Signs Of Circular Functions.......................102

Fundamental Trigonometric identities 103

Circular Functions of Special Angles .. .. 103
Angle of Elevation/Angle of Depression .........e.................... .. 104
Calculus ......................105
TRIGONOMETRY TEST A 106
TRIGONOMETRY TEST B ...... . .............................. 109
WORD PROBLEMS . ........................112
Number Relation Problems.....................................v........................... ............112
Investment Problems
113
Coin Problems . .............................114
Mixture Problems .. .........115
Motion Problems ...........................119
Work Problems 120
Age . .............. ..............122
PROBLEM SOLVING EXERCISES A ......124
PROBLEM SOLVING EXERCISES B .. ..............126

Mathematics 2014 09155057703 3998584

 PTS Tutorial and Review Systems Katjpunan Branch 2014

ALGEBRA OF NUMBERS AND INTEGERS

REAL NUMBER SYSTEM
A real number system is a positive, negative, or zero and can be classified as either rational or irrational number.
Notations:
R set of real numbers
Q set of rational numbers Q' set of
irrational numbers z set of integers {...-
2, -1, O, 1, 2,...} set of positive
integers

Real Number System

PROPERTIES OF REAL NUMBERS

For any real numbers a, b, c, we have the following
PROPERTY ADDITION MULTIPLICATION
CLOSURE
COMMUTATIVE ab = ba
a+b=b+a
ASSOCIATIVE a(bc) = (ab)c
DISTRIBUTIVE a(b + c) — — ab + ac
(a + b)c = ac + bc

IDENTITY
INVERSE 1
a
41page
BASIC OPERATIONS OF REAL NUMBERS AND SUMMARY OF KEY CONCEPTS
Positive and Negative Numbers A number is denoted as positive if it is directly preceded by a "+1' sign or no sign at
all. A number is denoted as negative if it is directly preceded by a - sign.
Opposites Opposites are numbers that are the same distance from zero on the number line but have opposite signs.
Double-Negative Property —(-a)=a
Absolute Value (Geometric) The absolute value of a number a, denoted lab is the distance from a to O on the
number line.
Absolute Value (Algebraic)
The absolute value of a real number x is denoted by Ixl. This is defined as:
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x, if x > 0 Ixl
= —x, if x < 0
o, if x = 0
Example:
1. 1—31 = 3
1 1 2.
2 2
3. Solve for x in Ix — 41 = 6
Solution CASE

x —4=6
x = 10
CASE 2: x —4 < 0

4. Solve for x in Ix+Sl

Solution
Since absolute value of a number is never negative, then Ix + 51 = —3 has no solution
Addition with O
0+any number—that particular number, that is, 0+a=a for any real number a.
Additive Identity
Since adding O to a real number leaves that number unchanged, O is called the additive identity.
Reciprocals
Two numbers are reciprocals of each other if their product is I, The numbers 4 and are reciprocals since

Negative Exponents
1
If n is any natural number and x is any nonzero real number, then x —

LAWS ON SIGNED NUMBERS

Law 1 To add two real numbers with the same signs, add the numbers and copy the sign.
3+5 8
Law 2 To add two real numbers with different signs, subtract the lower number from the larger number disregarding
the signs, then copy the sign of the larger number for the result.

Law 3 To subtract two real numbers, change the sign of the subtrahend and proceed to algebraic addition
9 — 16 = 9 + (—16) — —9 — (—16) = —9 + 16
—9 16 = —9 + (—16)—25
Law 4 To multiply (divide) two numbers having same signs, multiply (divide) the two numbers and the sign of the
result is always positive. 4(6) 24
24
Law S To multiply (divide) two numbers having different signs, simply multiply (divide) the two numbers amd the
sign of the result is always negative.
-8(2) = —16 - —16

2 —2
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Performing Operations with Real Numbers

The four basic operations of mathematics are addition, subtraction, multiplication, and division. The primary rule in
performing operations is summarized in the mnemonic PEMDAS which stands for Parenthesis, Exponents,
Multiplication or Division, Addition or Subtraction.

PE MDAS
Parenthesis
Multiplication
Division Addition
Exponents Subtraction
Examples
1. (42 2) + (6+ 8 + 2)
(16-2 — 2) + (6+4)
(32 - 2) + 10
30+ 10
3
2. —9 2 (—3)Ø— 58 —2•
—81-: — 29
9 29 —20

3.
—30+ 2
-15
- 2] + [-42 -1 2 ]
-4) - 2] + [-16 - 1]
1—36 + 4 — 2] * —[17]
34 ¯ 17
2
 PTS Tutorial and Review Systems Katipunan Branch 2014

DECIMAL, FRACTIONS AND PERCENT

Operations with Decimals
Add: 32.625, 6.215, 521.8436, 6
32.6250
6.2150
521.8436
6.0000

566.6836
Subtract: 32.625 from 41
41.000
—32.625

8.375
Operations with Fractions
Rule 1 To add and subtract fractions with same denominator, add the numerator and copy the denominator
Example

2.
Rule 2 To add and subtract fractions with different denominators, first, we would like to convert the fractions into a
form where all the denominators are the same. To do this, find the least common denominator (LCD). Divide the LCD
by the denominator of the first fraction then multiply the result to its numerator. Do the same for all other fractions
in the equation until they all have the same denominator. Then follow rule number 1 to solve.
Example
4 (2)+3 (5) 8 + 15 23
12 12
2. 8-15

12 12
Rule 3 To multiply fractions, multiply the numerators, then multiply
the denominators. Example

1. 3(5)
2.
Rule 4 To divide fractions, multiply the dividend to the reciprocal of the divisor. Example

2.
note: convert mixed numbers to improper fractions
Rules to Follow in Changing to Decimals
A. For Fractions: Divide the numerator by the denominator using the long division Example:
1. 0.75
4
2. 0.33
3
B. For Percentages
Changing Percent to Decimals
1. Divide the percent by 100% SAMPLES:
75%
75% 0.75 — = = = 0.75
100% 100
345%
— = = 3.45 100% 100
 PTS Tutorial and Review Systems Katipunan Branch 2014

2. Move the decimal two places to the left and omit the percent sign
SAMPLES:
25% = 0.25 7% = 0.07
Changing Decimals to Percent
1. Multiply the number by 100 then add the % sign
SAMPLES:
3 = 300% =
300%
.621 62.1% .621x100%62.1
2
= 50%
1
= 50%
2 2
2. Move decimal point two places to the right hen put a % sign
SAMPLES:
0.25 = 25% 0.07 = 7%
Changing decimals to Fractions A.
Terminating decimals
SAMPLES:
6 3
1. 0.6 =
10 5

3. 4.75 = 4+ 4 +2 = 420r 19

100 4 4 4
B. Repeating Non-Terminating Decimals
SAMPLE: 0.333 — o.
3
SOLUTION: Let x = 0.33 ...
lox = 3.33 . . .

Therefore 10x = 3.33 ... -x =

0.33 ...

g 3
1
Thus 0.333 ... —
3
EXERCISES: Express the following decimals in fractions
1. 1.9 2.3. 3.1.4f4
J
NOTE:
1. In changing fractions to decimals, divide the denominator to the numerator using long division.
2. In changing percentages to decimals, divide the percent by 100%
3. In changing decimals to percent, multiply the number by 100%
SCIENTIFIC NOTATION
This involves powers of 10. For any positive number x, it can be written as
follows. x = ax IO C where 1 S a < 10 and cis an integer
Changing Number to Scientific Notation
When a number is expressed in this form, it is said to be written in scientific notation. SAMPLES:
 PTS Tutorial and Review Systems Katipunan Branch 2014
PTS Tutorial and Review Systems Katipunan Branch 2014

Changing Scientific Notation to number (Standard Form)

Move the decimal point in the first factor corresponding to places indicated by the exponent for
the power of 10. The decimal point is moved to the right if the exponent is positive and to the left if the
exponent is negative.
SAMPLES:
1. = 46,210
2. 3.695x10-2 = 0.03695
RATIO AND PROPORTION
When two ratios are given, they can be written in fractions, then use cross multiplication to get
the result. Any fraction can be described as a ratio.
Proportion is an equality of two ratios. It is denoted by a : b = c d read as 'la is to b as c is to d", that is = —
where a and d are called extremes and b and c are called means.
SAMPLES:
1. Which of the following is/ are proportion?
8:10
ll.
Ill.
The answer is Il
— 5(8)
2. Express 27: 81 in lowest term
SOLUTION: 27: 81 27 27
1

81 27 3
27:81
Therefore,
=
3.
Find the value of x in 6 96
SOLUTION:
6 x
8 96
8x — 6(96) 8x = 576 x =
72
4. The sides of a triangle are 6, 8, and 15 inches long. In similar triangle, the longest side is 21 inches long. Find the
shortest side.
EQUATION:6: 15 = x: 21
42
ANSWER: —
5
5. Two brothers are respectively 5 and 8 years old. In how many years will their ages be in ratio 4: 5?

ANSWER: 7 years

VARIATIONS
Kinds of Variation
1. Direct Variation If y varies directly with x or if y is proportional to x, then y = kx or k = —, where k is called
constant of proportionality.
2. Inverse variation
 PTS Tutorial and Review Systems Katipunan Branch 2014

If y varies inversely as x or if y is inversely proportional to x, then y = — or k= xy.

3. Joint variation
If y varies jointly as x and z, then y = kxz or k = 2-
xz
SAMPLES:
1. If z varies directly as x and inversely as y and if z = 10 when x= 5 and y = 3.
find x when z = 12 and
SOLUTION: kx yz = — 3(10) = 22 = 6
5 5
If z = 12 and y = 4, then x is
4(12
k )6

2. If x is proportional to y and z 2 , and if x = 36 when y = 2 and z = 3, find x when y = 3 and

z
SOLUTION: x = kyz 2 x2 = •c = 22 = 2
18 When y = 3 and z = 4, x is

x = 96
3. If 2 men can repair 6 machines in 4 hrs, how many men are needed to repair 18 machines in 8 hrs?
SOLUTION:
Hence the required number of men is:
x=men
y=machines
z=hours
ky
z
2(4) 4

6 3
When z = 8 and y = 18, x is
24

= — = 3 men
8

STATISTICS
MEAN , MEDIAN AND MODE
Mean is the average of the data in a set. It is the sum of all data or elements divided by the number of data or
elements.
Median is the middle of all elements or data in a set arranged in descending or ascending order. Odd # of
elements = middle term
Even # of elements = average of two middle terms
Mode is the most frequently seen element in a set
SAMPLE: Ronalds grade in Math and science for 2 semester
are shown below:
Math:
Science:
Find the mean, median and mode of each subject.
SOLUTION:
a) For Math
Mean 80+83+ 85+ 85+ +86
11
 2014

= 85.18
Median = 85
Arrange the grades in descending order: 90, 87, 86, 85, 85, 85, 85, 83, 82, 80. Since there are 11
grades, the middle is in the 6 place, which is 85. Therefore the median is 85.
Mode is 85, since 85 appears most frequently.
b) For Science
Mean =
10
= 86.3
Median=86.S
Arrange the grades in descending order we have: 90, 89, 89, 89, 87, 86, 85, 84, 82, 82. Since the
number of grades is 10 (even), the middle in the 5 and 6 places. Therefore, the median is the average
of 87 and 86 which is 86.5
Mode is 89, since 89 appears the most number of times than the others.
PERMUTATIONS, COMBINATIONS AND PROBABILITY
In English, we use the word "combination" loosely, without thinking if the order of things is important.
"My fruit salad is a combination of apples, grapes and bananas 'J
We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or " grapes,
apples and bananas", it is the same fruit salad.
"The combination to the sage was 472"
Now we do care about the order. "742" would not work, nor would "247 11 . It has to be exactly 4-7-2
In Mathematics, we use more precise
language:
If the order doesn't matter, it is a
COMBINATION If the order does matter,
it is a PERMUTATION
PERMUTATION
A Permutation is an ordered Combination
There are three types of permutation problems you need to know
• Repetition is allowed: such as the lock example. It could be "333"
• No Repetition: for example, the first three people in a running race. You can't be first and second.
• No Repetition and objects are not distinct: how do you solve for " How many permutations are there in the word
LOBBY?
Permutations with Repetition
When you have n things to choose from and you are choosing r of them, the permutations are: n
x n x (r times)
Example
In a combination lock, there are 10 numbers to choose from (O, 1, 2,
, 9) and you choose 3 of them
10 x = 1000 permutations
Page
The formula is:
n where n is the number of
things to choose from, and you choose r of them (repetition
allowed, order matters

Permutations without Repetition

Think about the following example:
In how many ways can 5 pool balls be selected out of 16 balls?
After choosing say number 15, you cannot choose it again, So your first choice would have 16 possibilities,
and your next choice will have 15 possibilities, then 14. And the total permutations would be
16 x 15 x 3360
The number of permutations of n distinct objects taken rat a time is

n Pr-P(n, r) = — — 2) ... (n —r + 1) —
SAMPLE: How many ways can a 3-digit number be arranged to form a bank password
SOLUTION: There are 10 digits taken 3 at a time

3)!
Permutation of n objects non—distinct objects
The permutation of n objects taken n at a time, in which q are alike, r are alike and so on is
P(n, r) =
SAMPLE: How many permutations are there in the word
LOBBY?
SOLUTION: Letter B = 2 Letter L = 1
Letter O = I Letter Y = I

n! 5! 5 - 4 - 3 - 2'
= 60 ways
Page
Combination
A combination is an unordered arrangement of objects in a set. The numbers of a combination of n objects
taken r at a time is n! C(n,r) =
SAMPLE:
How many combinations can be made on 6 girls taken 2 at a time to appear in a variety show?
SOLUTION:

= 15
Circular Permutation
How can A, B, C be arranged around a circle?

Circular Permutaüons
Not in the three ways shown above! Why? Because each one of A, B, C has the same neighbor. Without changing the
neighbor, only changing seats will not change the circular permutation. The only valid circular permutation is
as 0follows 0
Grcular Permutations
The formula is (n-l)!
n
Think about this example:
10 beads will be arranged to make a bracelet. How many possible combinations can be made?
Probability
Many events cannot be predicted with total certainty, The best we can say is how likely they are to
happen using the idea of probability.
If an event can happen in m ways and may fail in n ways, then the probability that it will happen is
And the probability that it will fail is

SAMPLE:
In tossing a coin, what is the probability that the head, H, will appear?
SOLUTION:
When a coin is tossed, there are two possible outcomes, heads(H) or tails (T).
1
2
SAMPLE:
When rolling a dice, what is the probability that a "4" will appear?
SOLUTION:
When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6
The probability of any one of them is 1/6
k.

NUMBER SERIES AND PROGRESSIONS

Sometimes, we can easily see the definite pattern in a number series.
A. Addition and Subtraction
1. In the series 8 16 24 32. The next number is 4(), since the pattern is adding 8 to the preceding number.
2. In the series 21 18 15 12 the next is 9, since the pattern is subtracting 3 from the preceding number, so
12-3=9.
B. Multiplication and Division
1. In the series 3 9 27 81 the next number us 243 since the pattern is multiplying 3 with the preceding
number.
2. In the series 16 8 4 2 the next number is 1, since the pattern is dividing the preceding number by 2
C. Other Number Series
1. In the series 2 6 14 30 the next number is 62, since the pattern is a combination of subtraction,
multiplication and addition, that is 6 —2 = 4x 2 = 8 + 6 = 14, then 30— 14 = 16 x 2 = 32
Progression
A. Arithmetic Progression
An arithmetic progression is a sequence of numbers in which each term, after the first is obtained by adding
a fixed number to the preceding term called the common difference.
The following are examples of arithmetic progressions with the computed common difference d:

+ 4p
If a denotes the first term of an arithmetic progression and d for the common difference, then the n terms
of the arithmetic progression are a, a + d, a + 2d,a + 3d
I)d
From the foregoing elements the nth term,denoted by al,of an arithmetic progression is an =
al + (n — l)d
SAMPLE: Find the 30th term of the arithmetic progression. 10, 7,
SOLUTION:
Given d = 7 -10 -- -3, al = 10, and n = 30. Substituting these values in the formula an = al
+ (n — I)d
10 + (30 - = ¯ 77
SAMPLE: Which term of the sequence 4, 7, 10, is 52?
SOLUTION:
Given al —4, d = 7—4 = 3, and an =52 substituting these values in the formula for gives
52 = 4+(n n = 17
Sum of the terms of Arithmetic Progression
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SAMPLE; If a boy saved P25 on the first day, P27 the second day, P29 the third day, and so on, how many
days will it take him to save P880?
SAMPLE: A raffle ticket consists of 20 prizes. The first prize is P25,OOO, and each successive price is P500 less than
the preceding prize. What is the value of the 19 prize?
B. Geometric Progression
A geometric progression is a sequence of numbers in which each term, after the first, is obtained by
multiplying each preceding number by a constant called the common ratio. The elements of the sequence are
called terms.
The following are the notations used for geometric progression:
al = first term r = common
ratio n = number of terms
an = last term or n tern

The terms of the geometric progression may be presented by a, ar , ar 3, ..., ar

From the foregoing elements, the n term, denoted by an , of a geometric progression is
it—I

a = air
The following sequences are geometric progression:
1
2
9 1

1 1
31—
3
Sum of the First n terms of Geometric Progression al (1 —
rü)
s
I—r
SAMPLE: The third term of a geometric progression is 144 and the sixth term is 486. Find the first term of the
geometric progression.
SOLUTION:
The sixth and third terms of geometric progression are as follows:
= ars = 486
= = 144
5 3 27 3
By Division,
8 2
Then the first term is obtained by substituting — in a3 or a6 is:
9 al = 144 or, (11264
B.
Figure 2 — Number of jobs finished in 8 months
C. CIRCLE GRAPH/PIE CHART — the circle graph is used for comparison of quantities
represented by sectors of a circle.
Figure 3 — Percentage of employees

1. A close examination of the graph shows that the largest annual population growth was during the
period?
(A) 1978- 1979 (D) 1984 -
1985
(B) 1981-1982 (E) 1987 -1988
(c) 1983 -1984
2. Which of the following statements is true?
(A) Population education was most successful from 1985 — 1986
(B) Population education was a failure from 1985 — 1986 (C)
Population growth in the country is steadily decreasing.
(D) Population in the country is beyond control.
(E) Population in the country will drop by 1990
3. What was the population growth of the country from 1978 — 1979?
(A) 2.6 million (D) 3.5 million
 PTS Tutorial and Review Systems Katipunan Branch 2014

(B) 2.2 million (E) 3.7 million (C) 2.0 million

4. During what period does the graph show the lowest population growth?
(A) 1978- 1979 (D) 1987 - 1988
(B) 1990-1991 (E) 1983 -1984
(C) 1981-1982
5. By 1992 and after, which of the following can be deduced from the data
(A) Population will decrease (D) Population will double
(B) Population will increase (E) Population will stop growing
(C) Population will not change
|
Pie Chart of the Family
Monthly Budget
Miscellaneous
7%

1. The circle graph or pie chart shows the smallest portion of the family budget is?
(A) Food and Recreation
(B) Shelter, Clothing and Socials (C) Education, Transport, Health
(D) Miscellaneous
(E) Savings and Insurance
2. What item in the budget is about quarter of the monthly income fo the family?
(A) Shelter , Clothing and Socials
(B) Food and Recreation
(C) Education, Transport, Health
(D) Miscellaneous
(E) Savings and Insurance
3. If the family monthly income is PIO,OOO how much will be the appropriated for shelter, clothing and
social?
(A) P2,700.OO (D) P2,OOO.OO
(B) P2,500.OO (E) PI,800.OO
(C) P2,200.OO
4. If the family monthly budget is P30,000 how much is the difference of the budget for Education,
Transport, Health than Shelter, Clothing and Socials?
 PTS Tutorial and Review Systems Katipunan Branch 2014

(A) 3000 (D) 3850

(B) 5000 (E) 4000
(C) 8000

1. During the first six months of the year, which month had the highest budget?
(A) April (D) June
(B) March (E) December (C) May
2. During what month is the budget during the year the lowest?
(A) September (D) April
(B) May (E) December (C) January
3. What is the approximate highest budget of the school during the year?
(A) 650M pesos (D) 700M pesos
(B) 450M pesos (E) 750M pesos (C) 250M pesos
4. How many times in a month does the budget appear at about 600 million?
(A) 2 times (D) 5 times
(B) 3 times (E) once
(C) 4 times
24 1

BASIC MATH TEST A

Algebra of numbers and integers
1. What property is illustrated by the fact that — PW + 49W
(A) Associative property of Multiplication
(B) Commutative property for addition
(C) Distributive property
(D) Identity property of Multiplication
2. Arrange the numbers in correct order: 6.3, T, —7.8,

(A) 6.3 < < —7.8

(B) —7.8 < 316 < T < 6.3
(C) NIG > 6.3 > T > -7.8 (D) —7.8 > > 6.3
3.
(A) 438 (C) 62
(B) 342 (D)
2582
 4. Which of the following cannot describe an integer x?
(A) It is a prime number.
(B) It is an irrational number (C) It is not a prime number (D) It is an even number.
5. A deep-sea diver rises to the surface in two steps of 18 feet each. How far did the diver
rise?
(A) 18 + 2 = 20, an ascent of 20 ft
(B) 18- 2 = 16, an ascent of 16 ft
(C) 18 +2 = 9, an ascent of 9 ft
(D) 18 • 2 = 36, an ascent of 36 ft
Decimals and Fractions
6. Evaluate: (1.75y + (2.1Y =
(A) 123.235
(B) 12.3235
(c) 1232.35
(D) 12323.50
7. Which of the following sets are in ascending order?
111 121
(A) (c)
F'i'ä
4'
3 111
(B) (D)
3'4'2
8. Express 0.6 in fractions
2 1
3 3
3
(B)
5
9. Ano ang kalahati ng lima na idinagdag sa kaapat ng labing anim?
4 13
10 2
41 9
10 2
10. What part of an hour elapsed between 1: 56pm and 2:16pm during a certain daytime?

(A) 0.27 (C)

0.15
(B) 15 (D) 1.5
 PTS Tutorial and Review Systems Katipunan Branch 2014

4 1
15 3
1 8
(B)
6 15
Percent
11. What percent of 80 is 30?

(A) 357% (C) 260 2

%
2
(B) 37 % (D) 26
2
12. What is-of 75%? %

13. If a side of a smaller square is 1.6cm and a side of the larger square is 2.4cm, how much bigger is the area of
the larger square relative to the area of the smaller square in terms of percent?
(A) 125% (C) 225%
(B) 175% (D) 275%
Scientific Notation
14. Express 0.00605 in scientific notation
(A) 0.065 x 10 -3
(B) 605 x 10 -5
(C) 6.05x 10-3
(D) 6.05x 103
15. The number of DVDs sold by one company in 2010 was approximately 2.0 x 10 6 . The number of DVDs sold
by the same company in 2012 was approximately 9.2 x 10 7 . How many more DVDs were sold in 2012 than in 2010?

Ratio and Proportion

16. There is a law stating that the ratio of the width to length for the American flag should be (10:20) in. Which
one of the following dimension is in the correct ratio?
(A) 30 in by 60 in (B) 80 ft by 152 ft
(C) 40 ft by 76 ft
(D)80 in by 100 in
17. Write the ratio as a rate on lowest terms: 416 kilometer in 8 hours?
(A) 47 km/h
(B) 52 km/h
(C) 57 km/h
(D) 62 km/h
18. Four pounds of grass seed will cover 100 square feet. Let p represent the number of pounds of seed needed
to cover 450 square feet. Select the correct statement of the given condition.
(A) 22.2
450
4 450

100 p
100 _ p
450 4
4 p
100 450
 PTS Tutorial and Review Systems Katipunan Branch 2014

19. The figure below show two circles. Circle a has a radius of 9 and circle b has a radius of 3. What is the ratio
of the area of the smaller circle to that of the larger?
(A) 1:81
(D) 81:1
Variations

20. Find x if y varies inversely as twice x and y is 5 when x is 1

5
3

5
21. If z is 5 and y is 4, then x is 80. If x is jointly proportional to z and y, find z if x is 20 and y is 5.
(c) 10
22. Quarter of x changes directly as 5y. When y is 5, x is 100. Find y if x is 60.

Mean, Median, Mode

23. The average of three numbers is 12. What is the third number if the average of the first two numbers is 10?
(A) 16
(B) 22
(c) 26
24. Si Mhy ay nagkamit ng markang 78 sa English, 86 sa Biology, at 82 sa Filipino. Ano ang kailangan niyang
marka sa Math kung ang dapat niyang average sa apat na subjects ay 85?
(A) 92 (C) 94
(B) 93 (D) 95
25. Find the average price for the following catalog items.
Price Number of Items
$11 4
$14 2
$17 7
$20 1
(A) $14.07
(B) $15.07
(C) $15.50
(D) $52.75
26. Find the median for each set of data:
(A) 21
(D) 10
27. The pulse rates of students before physical education class are recorded in the table.
Pulse Rate 68 69 70 71 72
No. of Students 3 2 2
Pulse Rate 73 74 75 76 77
No. of Student 1 2 o 5
1
What is the mode for the pulse rates?
 (A) 76 (C) 70
(B) 77 (D) 74
Permutation, Combination, Probability
10!
28. Gawin sa simpleng porma ang —
7!
(c) 27
(B) 90 (D) 720
29. A card is drawn from a standard deck of cards. Find the probability that the card is either a nine or a black
card.
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PTS Tutorial and Review Systems Katipunan Branch 2014

Sequence/ Number Series

30. Find the next number of the series 12, 14, 17, 22, 30, .
(A) 45
(B) 44
(C) 42
(D) 41
31. Find the 7th term of the geometric sequence for the al = —3 and r — 1
1
729
3
1

3
32. Find the common difference of the arithmetic sequence: 1.9, 2.4, 2.9, 3.4
(A) 0.4
(B) -0.5
(c) -0.4
(D) 0.5
33. Find anof an arithmetic progression if the first term is 8 and the common difference is - 0 6
(A) an = 7.4 + O.6n
(B) an = 7.4 - O.6n
35. The graph (C) an 8.0 — 0.6n
(D) = 8.6 — o.6n
2001-2006.
Data Interpretation
(A)
34. A radio station asked listeners to vote for their favorite type of music. They displayed the
(B)
results using the double bar graph shown below. Which music is most popular overall?
(C) (A) Rock
(D) (B) Country
36. The graph (C) Oldies
dealer for the (D) Pop
sedans were
/
(A) 120%
(B) 7
5
%
(c)
37. In a young
100%
210 is spent
(D)
50%
(A)

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BASIC MATH TEST B

Algebra Of Number
1. If (a + b) is negative, then la + bl =

2. Identify the graph that shows the correct plot of the point: IT, 10, —6, MS

(B)

(C)

(D)
3. Simplify: (—5 — (A) 38
(B) -34
(c) -54
(D) -16
4. Leo drank 2,126 quartz of milk. Jay drank 3,104 quartz of milk. How much more did Jay drink than
Leo?
(A) 1082 (C) 1088
(B) 978 (D) 1078
5. Si Marison ay nakatira na may 30km ang layo sa eskuwelahan. Magkanu ang ibabayad nya sa driver
ng jeep papuntang eskuwelahan kung ang pamasahe ay P6 sa bawat 3km?
(A) P60 (C) P18
(B) P30 (D) P15
Decimals and Fractions
6. Which of the following has the smallest value?
(A) -0.02
(B) -1.03
(C) -2.52
(D) 0.009

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7. Simplify: 1
8. Express I. fraction.
17 99199
1
11
9. Which of the following variable when double can half the value of •
2b+4(c-d)
(C) c and d
(D) none of the above
10. Martha prepares a baked macaroni which weighs 1 kg. After putting it in a microwave oven, — of its
contents burned. What is the weight of the baked macaroni that Martha can serve?
(A) kg (C) kg
(D) kg
Percent
Il. If of a certain number is 56, then one-half of that number is
4
(C) 112
(B) 11,200 (D) 0.07
12. When the minimum fare of a passenger jeepney from UP Campus to SM North is increased for P8.OO to PIO.OO.
What is the rate of increase?
(C) 25%
(D) 169/0
13. What is— of 1%?
3 (c) 0.03
(B) 0.3 (D) o.ooä
Scientific Notation
14- Write in decimal notation:123x106
(A) 123,000

15. Last year a large trucking company delivered about 9.6 million tons of goods at an average value of $30,000
per ton. What was the total value of goods delivered? Express your answer in scientific notation.
(A) $96.OX1010 (C) $28.8x10 10
(B) $9.6x10 11 (D) $2.88x10 11
Ratio and Proportion
16. A math review class consists of 15 males and 12 females. Find the ratio of males to the entire
class.
(A) 15:12 (C) 12:15
(B) 15:27 (D) 12:27
a
4
17. If-=
b

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Mathematics 2014 09155057703 / 3998584
 (D) none of the above
18. There are 5 male students for every 10 female students at a local school. Select the statement of the
condition when there are 23 male students
23 5 23 s x
10 x 23 10 (D) none of these
5 x
19. Ang kahon ay may lamang bola na pangbasketball at pang-volleyball. Kung ang ratio ng bola ng basket ball
sa bola ng volleyball ay 3:7, at kung may 28 na bola ng volleyball, ilang bola ng basketball ang nasa kahon?
(A) 12 (C) 62
(B) 15 (D) 18
Variations
20. If x = 30 when y = 50 and x varies directly with y, then find x when y=190
(A) 119 (c) 124
(B) 114 (D) 104
21. The square of x is jointly proportional to the cube of y and multiplicative inverse of z. If y is 1 and z is 3
whenever x is 2, then what is the value of z if x is 3 and y is -?
3
22- Rica's salary changes directly as the company's gross income. If the company has an income of PIM /month, Rica
receives PI,OOO. How much was the company's income if Rica received P15,OOO?
(A) P15,OOO
(B) P150,OOO
Mean, Median, and Mode
23. Mike was in charge of collecting contributions for the Food Bank. He received contributions of $40, $50,
$40,
$80 and $20 from five coworkers. Find the median value of these contributions.
(A) $45 (c) $46
(B) $60 (D) $40
24. In three compartments, there were 12 bundles, 13 bundles and 8 bundles,
respectively. If the bundles were rearranged so that each compartment held the
same number of bundles, how many bundles would be in each compartment?
(A) 11
(B) 12 (D)IO
25. Mr. Farquand recorded the number of sick days each employee had last
year, as shown in the table.
EMPLOYEES SICK
DAYS
Earl 6
Hayden
1
Sarita 7
Ryan 6
Camille 2
Isabel 4

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Javier 3
Marty 4
Mariko 3
Which of the following represents the mean number of days
employees were sick? (A) 3.21
(B) 4.24 (D) 4.79
26. The average of the ages of Danny and Jami is 27. If Jami is 12 years old,
then how old js Danny?
(A) 42 (C) 15
(B) 29 (D) 52
27. The pulse rates of students before physical education class are
recorded in the table.
Pulse 68 69 70 71 72 73 74 75 76 77
Rate
No. of 2 5 2 3 3 3 2 3
Students 1
What is the mode for the pulse rates?
(A) 73 (c) 69
(B) 75 (D) 68
Permutation, Combination and probability
28. Evaluate —
(A) 34 (C) 60
(B) 10 (D) 12
29. Account numbers for Western Oil Company consists often digits. If the
first digit cannot be a O or 1, how many account numbers are possible?

Sequences and Numbers Series

30. Find the next number of the series 27, 32, 37, 42,...
(A) 47 (c) 46
(B) 45 (D) 48

1 1 1
31. Identify the missing term in the following geometric sequence: -2.1. —
8'
32
1
16
32. For the arithmetic progression 7, 4, 1,.... Find ag.
(A) -24 (c) -17
(B) -23 (D) -20
33. Find the sixth term in the sequence.• 6, 10, 14, ....
(A) 26 (C) 30
(B) 20 D) 34
Data Interpretation
34. A radio station asked listeners to vote for their favorite type of music.
They displayed the result using the double bar graph shown below. Which music
is most popular with the women?

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(A)
Country (C) Pop
(B)
Rock (D) Oldies
35. Tenth-grade students were surveyed to determine how many hours per day
they spent on various activities. The results are shown in the circle graph
below. About how much time altogether was spent on sleeping and watching TV?
(A) 11 h (C) 16 h
(D) 46 h
1396M HOW Students Spend Their Time
Watching TV
Soctakzing
33
Sleeping
Eating
36. The graph below represents the annual rainfall in inches for 2005-2010. How
much did it rain in 2006?

(A) 10 in (C) 30 in
(B) 35 in (D) 15 in
37. According to the circle graph, which is the largest source air pollution?
Soarces of PoUon
Buses and Trucks
25 %
Autos
35 %

Industry

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(A) Industry (C) Buses and Trucks

(B) Autos (D) none of the above
DATA INTERPRETATION EXERCISES
Figure 1

1. Who was the employee who planted the most number of trees?

2. Find the average number of trees planted by all six employees

3. Who was the employee who planted more than the average with the least difference?

5. What percent of the number of trees planted by employee number 5 is the number of trees planted by
employee number I?

6. Who was the employee who planted 30 trees?

7. What was the difference of employee 1 and 4 in terms of trees planted?

8. How many trees were planted by all the employee?

Figure 2
Circle Graph of an Organization's budget for different activities

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Circle Graph of an Organization's Budget

for Different
Activities
E Miscella
neous
Extension
Production
Research
Instruction

1. What is the first priority of the organization?

2. What activity is one-fourth of total activities?

3. What amount of the total budget will be for instruction if the total budget is P 800,000?

4- If the budget for Research is P 100,000 how much would the total budget be?
5. What is the difference of the budget for Research and Instruction if the total budget is P

6. If the total budget is P 80,000 how much will be for Miscellaneous?

7.What is ratio of Production to Extension activity?

8. What is the difference of the budget for Production and Miscellaneous if the total budget is P 300,000?

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Figure 3 Line Graph of College Enrollment Data

I. What year did the college have the highest enrolment?

2. What year did the college have the lowest enrolment?

3. What was the trend of enrolment in the college after year 2006?

4. What was the difference of enrolment between 2006 and 2010?

5. What was the ratio of the enrolment of the 2006 and 2009?

6. What was the enrolment in 2007?

7. What was the average enrolment during the 5-years period?

8. If the enrolment in 2006 is 100, what was the enrolment in 2009?

9. What was the lowest decrease in enrolment?

10. At what year does the average enrolment fall?

ALGEBRA
POLYNOMIALS/ALGEBRAIC EXPRESSIONS
(A) 3x2 +2x is a polynomial in x
(B) —2x 3y y is a polynomial in x and y
20abc .
IS a polynomial in a, b and c
(D) ts a polynomial in x and y but not in z
2
(E) (8x)ä is not a polynomial since 8x has non-integral exponents.
(F) —x¯ is a monomial but is not a polynomial in x because of the negative exponent of x
Degree of a Term: Degree of a Polynomial
The degree of term in a particular variable is the exponent of that variable.
The degree of a term in two or more variables is the sum of the exponents of the variable. The
degree Of a polynomial is the degree of its highest-degree term or terms in a variable.

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SAMPLE: The degree of —2x 2y4 is 2 in x and 4 in y. However, the degree of —2x 2y4 is 6 in x and y.
O erations with POI nomiats
Polynomials or algebraic expressions are written using grouping symbols. The basic grouping symbols are
parenthesis, brackets and braces.
Addition and Subtraction
To add or subtract polynomials, simplify the given algebraic expression by removing grouping symbols and
combining similar terms.
SAMPLE:
2aL3a(a + I) + (—2a 2)]
= 2a[3a2 + 3a — 2a2]
= 2a[a2 + 3a] = 2a3 + 6a2
Law of Exponents
1. Product of Power a m a n = a m +n
2. Quotient of Powers a
In—n an

3. Power of Power (am)n = amn

4. Power of a Product (ab)m = ant b tn
5. Power of a Quotient
6. Negative Exponent
7. Zero Exponent
Multiplication and Division
To multiply polynomials by another polynomial simply get the product of all numerical coefficients and the
product of all literal coefficients. Apply the laws of exponents.
To divide monomials by another monomial use the exponential rules. To divide a polynomial by a monomial,
divide each term of the polynomial by the monomial.
SAMPLE:
[5(x — — y)2 1 = 40(x —
Division of Two
Polynomials
To divide two polynomials, perform the following steps:
1. Arrange the terms of the dividend and divisors in descending powers of
the variable. The missing powers are filled up using O as a numerical
coefficient of the term.
2. Divide the first term of the dividend by the first term of the
divisor to get the first the term of the quotient. 3. Multiply the
first term of the quotient to each term of the divisor and subtract the
result from the dividend. The difference will be the new dividend
4. Repeat step 2 and 3 to get the succeeding terms of the quotient. The
process is terminated only when the power of the new dividend is less than of
the first term of the divisor.
5. Express the result as:
— = Q + — where P is the dividend, D the divisor, Q the quotient and R the
remainder.
This can also be written as P=QD+R
SAMPLE:
Divide 2x 3 — 9x 2 + 6x — 5 by
2x — 1 ANSWER:
2x3 9x 2
+ 6x — 5 4
= x2 — 4x + 1 —

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Synthetic Division
Synthetic Division is used only when the divisor is of the form (x — r)
or (x + r)
NOTE: The polynomial terms must be arranged in descending powers. In case there
are missing powers, a zero coefficient must be used.
SAMPLE:
Find the quotient: (12 + x 2
— 7x)
SOLUTION:
Set all the numerical coefficients of the dividend in descending powers. That
is-
2 x x x 1
12 3
3
-12
12+xz-7x
Therefore, = —4,
remainder is 0 x—3
SPECIAL PRODUCTS AND FACTORING
The easiest way to find the product of polynomials is by the method
called Special Product Formula. The following are the different kinds of
special products.
1. Product of Two Binomials
(ax + by) (cx + dy) = acx2 + (ad + bc)xy +
bdy2
2. Square of Binomials
= X2 + 2XY Y 2

— x —2xy+Y2
/
3. Product of the Sum and Difference of the same two numbers (x + — y) = x2 —
Y2
4. Cube of a Binomial x3 + 3x2y + 3xy2 + Y3
(x —y) = x3 — 3x 2y + 3xy2 — y 3
5. Special case of Product of Binomial and Trinomial
(x + xy + y 2) = x3 + y3 (x + + xy+ y 2 ) =
x 3 —y 3
6. Square of a Trinomial
(a b + c) 2 = a2 + b2 + c2 -l- 2ab + 2ac + 2bC
Binomial Expansion

Higher power of the binomial (X+Y) may be derived by using Pascal Triangle as given below. This is used in
determining the coefficients of the expansion.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
17 21 35 35 21 7 1 and so on..
Let us relate the numbers of the Pascal Triangle with the coefficient of the expansion of (x +

x + 2xy + Y2
= x + 312 y + 3XY2 + y3
5x y
 + + + y4
4
+ 10x 3y 2 + 10x 2 y3 + 5xy4 +

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FACTORING
Factoring polynomials is the reverse process of the special product polynomials.
Types of Factoring

1- Common Monomial Factor ax + ay = a (x + y)

SAMPLE: 18a3 b4w — 12ab 2
ANSWER: 6ab 2 (3a2 b 2 w — 2)

2, Difference of Two Squares x2 —Y 2 = (x + —y)

SAMPLE: 4x 2 — 49y1
ANSWER: (2x + 7y 3 ) (2x - 7y 3 )

3. Sum and Difference of Two Cubes x3 + y 3 = (x

+ — xy + y 2) x3 y3 = (x + xy + Y2)
SAMPLE: 64x4 y — xy 4
ANSWER: xy(4x — y)(16x 2 + 4xy + y2 )

4. Perfect Square Trinomial x 2 + 2xy + Y 2 = (x +

2
x — 2xy + y
SAMPLE: 4x2 12 xy + 9y2 ANSWER: (2x — 3y)2

S. Other Trinomial (Trial and Error Method) x 2 + (a+b)x + ab = (x

+ a)(x + b)
SAMPLE: x 2 5x + 6
ANSWER: (x — 2)(x — 3)
SAMPLE: —1
ANSWER: (x + 1) 2 (x— 1)
RATIONAL EXPRESSION
Rational expression or a fraction is a quotient of two algebraic expressions or polynomials.
A fraction is said to be at its simplest form if the numerator and denominator have no common factor
except ± I.
SAMPLE:
(1 —x)(l + x) x 2 — (x x)
2x+1¯ (x — — 1)
Addition and Subtraction of Rational Expressions
SAMPLE: (Perform the operation and simplify)
1 2 3 x2—2 x(x—l)
x2—1 x(x+l)(x—l)

1 . Factoring by Grouping

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PTS Tutorial and Review Systems Katipunan Branch 2014

Multiplication and Division of Rational Expression To

multiply and divide two fractions, we have
ac
ad
a c ad
SAMPLE:
X -1-3 x—5

Find the product of and x2—9x+20

X2 +8X+15
1
ANSWER:
COMPLEX FRACTION
Complex Fraction is a fraction whose numerator or denominator or both contain fractions.
SAMPLE:

Simpli
fy

x
1
ANSWER:
RADICALS
1. Negative Exponents
Samples: Simplify the following without negative exponents:
-1

-21

3+2

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ll. Fractional Exponents
The form = a7 is called the principal nth rootofa m . The numerator m indicates a power and
the denominator n is called the root of order.
Specifically, —7 means the principal root of a. The symbol is called a radical, where a is called the radicand,
and n is called the index or order.
To evaluate radicals, it is sometimes convenient to express the radical with fractional exponent
or apply the rule = a.
SAMPLE:

2x2 (2x 2 ) 4x4

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Three Ways to Simplify Radicals

1. Removal of perfect nth powers
Break down the radicand into perfect and non perfect nth powers and apply the property =
SAMPLE:
Perfect owers non perfect owers
a. 3 32x4y32 3
= 2xy10

= 2a2b25
2. Reducing the index to the lowest possible order
Reducing the index is done by applying the property
SAMPLE:

3. Rationalizing the denominator of the radicand

Rationalizing the denominator of the radicand is the method used to remove the radical sign in the
denominator.
SAMPLE:
3
3 3 2 18

a. 3xy3 34X4y2
3
S
581X4y2
b.
3xy
Ill. Addition and Subtraction of Radica Is
Similar radicals are radicals of the same order and the same radicand. Similar radical can be combined
into a single radical by the use of the distributive law. Radicals of different indices and different radicands are
called dissimilar radicals.
This illustrates the addition of radicals:

IV. Multiplication of Radicals

• To Multiply two radicals of the same order, use the law on radicals

SAMPLE:

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To Multiply two radical of different order, it is necessary to express them as radicals of the same order.
SAMPLE:
= 2E 35
2 3

V. Division of Radicals To divide two radicals of the same order, use the third law on radicals and rationalize the
denominator of the radicand.

SAMPLE:
4
1 1 4427x2y
3x2y3 ¯ 3 3x 2y xy
To 3
3 3x 2y divide the radicals of different orders, it is necessary to express them as
radicals with the same order. SAMPLE:
6
432
2
To divide the radicals with denominator of the radicand consisting of at least two terms, again
rationalize the denominator. SAMPLE:

a —
b

LINEAR EQUATIONS
A linear function y = mx + ban * O or f(x) = mx + b gives a corresponding linear equation mx + b = O. The
root of this equation gives the zero at the function y = mx + b.
Many linear equation do not come in standard form mx + b = 0 like the equations below.
1. 3x-4 = 5- 2x
3x-4+ 2x-5 = o
9

2. x = —12

3. 3
LITERAL EQUATIONS
Sometimes you have a formula, such as something from geometry, and you need to solve for some
variable other than the "standard" one. For instance, the formula for the perimeter P of a square with sides of
length s is P = 4s. You might need to solve this equation for s, so you can plug in a perimeter and figure out the side
length.
This process of solving a formula for a given variable is called "solving literal equations". One of the
dictionary definitions of "literal" is "related to or being comprised of letters", and variables are sometimes referred
to as literals. So "solving literal equations" seems to be another way of saying "taking an equation with lots of
letters, and solving for one letter in particular."

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SAMPLE: (solve for the indicated

variable)
P = 21 + 2w,for I
21 + 2w = P
21 + 2m, -2w = P — 2w
21 = P — 2w
ANSWER: 1 =
2
SYSTEM OF LINEAR EQUATION IN 2 VARIABLES
A system Of two linear equations in two variables translates graphically to two lines. The ordered pair (x,
y) is a solution of the system of linear equation if it satisfies the linear equations the linear equations. A unique
solution (x, y) for the system exists when these lines are district and intersecting. However, it is possible that the
lines are parallel. in which case, there is no solution.
Illustration.'
4x — 6y = 1
Hence the system has no solution. It is also possible that the lines coincide in: 2x Hence
many solutions exist.

SAMPLE: We verify that both these values

Find the solution: Use elimination or Substitution satisfy the equation hence are roots
2x — 4(1) of the quadratic equation
SAMPLE:
ANSWER. Find the solutions of 2x 2 — x— 6 =
QUADRATIC EQUATION 0
A quadratic equation in xis SOLUTION:
in standard form if it in the form ax2 + Factoring the left-hand side,
bx + c — Some quadratic 2x2
equations that are not in standard
Solving for x in the linear
form are: 1.
2 equations
. 2x+ 3 = o x-2=0
3
l. Solution of Quadratic Equation by Factoring 2
Let f (x) = 0 be a quadratic equation. If f(x) is Hence, the quadratic 2x2 -— x
factorable into linear expressions g(x) and h(x) (that is, f 6 = 0 has two solutions, x = 3and
(x) = g(x) • h(x)) then, g(x) • h(x) = 0. We know that if a x=2
product is zero, then g(x) = 0 or h(x) = 0. Since the two
equations are now linear we can find their roots.
To illustrate: 2
3
C
(x + 2)(x
h
— I) = 0 x e
+ 2 = O or c
x— I = O k
t. x = —2 :
or X = 1

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If x = 2
—— If x = 2, we have
we
have 2(
Hence, x = 2 is another solution of
2
the equation.
= 2+ 2— 6 MORE SAMPLES:
-20+5x 2 = o
3x(1 — x) =
2
(x —2) (x— 3) 12
= 6—6
3
ANS: ANS: x = ±
Hence x = is a root of the equation 2x —x — 6 = 02 ANS: 6, -1
ll. Solution of a Quadratic Equation by Quadratic Formula.
The solutions of the quadratic equation ax 2 + bx + c , where a * O are given by

2a
This is known as the quadratic formula.
SAMPLE:
Use the quadratic formula to find the roots.x 2 + x

2(1) 2 2
ANSWER:
Ill. Solution of Quadratic Equation by Completing the Square
SAMPLE:
Solve 2x2 + x — 3 = O by completing the square. SOLUTION:

2 16 2 16

2
IV. Nature of Roots
We study the nature of the roots of a quadratic equation ax 2 + bx + c = O where a 0. From the
quadratic formula:

20
Consider the expression D = b 2 - 4ac , called the discriminant.

The roots of the quadratic equation ax 2 + bx + c — — O are:

A. Real and equal if D O
B. Real and unequal if D > O C. Non-real if D < 0
In particular, if D is perfect square, the roots are rational. If D is not perfect square, the roots are irrational.
SAMPLES:
Describe the roots without solving for the equation.
1.x2 — 2x+3 = o
D = 4- 12 < O ; non-real roots
2.x2 — 5x + 6 = O
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D = 25 — 1; rational roots, unequal

3.4x 2 — 4x + 1 = 0
D = 16- 16 = 0; rational roots, equal
4.x2 + x— 1 = 0
D = 1 —(—4) = 5; irrational roots, unequal
Exercise: Get the roots of the equations above

INEQUALITIES
Inequalities are also called relations of order, with symbols of > or < . To help visualize these relations we use the
number line.
Positive, Negative, Non-negative
x

-3 -2 -1 0 1 2 3 4 5
x is positive means x > O or x is to right of O on the number line y is
negative means y < O or y is to the left of O on the number line z is non-
negative means z 0 or z cannot be to the left of O on number line a > b
means "a lies to the right of b" on the number line or a is greater than b
We now introduce some notations for intervals on the number line: (a)
Open Intervals

(a, co) = (xlx > a)

( {xlx < a)
= {xlb < x < a)
(b) Close Intervals

= {xlb S x S a)
(c) Half-open or half-Closed intervals

1. x > 2 and x > -2 means x €

{(2,00) n (-2,00)} = (2,00)
-4 0 2 4

2. x > 2 or x > -2 means x €

(2,co) u (-2,co) = (-2,00)
-4 4

Absolute Value Inequalities

The diagram below illustrates the difference between an absolute value expression and two absolute value
inequalities.

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As seen in the graph, the solution set has a negative side and a positive side. Look at the example
Example:
12x— 61 > 4
Positive Side Negative Side

2x 6 > 4
2x > 10 -2x+6>4

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Solution of linear Inequalities
SAMPLES: Find the solution set of the following:

1.
3
Answer: x > —
2
2. 12x I
12x— sx 3-1
2
7xS
—

7
Solution with absolute Values
SAMPLES:

1. 25

2-
—4

Case I: Case 2.

3. 12x—31
Case I. Case 2

u > 5+ 3 2x < 5 +3

RELATIONS AND FUNCTIONS

A relation is any set of ordered pairs (x, y) of real numbers. The set of values of x is called the domain,
denoted by D of relations, and the set of values of y is called the range, denoted by R of the relation.
SAMPLE:
Relation defined by f(x) = (2,3),(3,4)} has the domain D = and range R =
The function is a relation such that no two ordered pairs have the same first element. A function
may be denoted asy = f (x) which is read "f of x".
Two Methods of Defining a Relation
1. Listing of ordered pairs
2. Rule Form
The last method is usually associated with an equation.
For example, the relation f(x) = can be described in a rule
form as f(x) = = x + I, x E Z)

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We can determine whether a given graph is a function using the vertical line test. If a vertical line intersects
in one point of the graplm the graph is said to be function.

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Let b = a x be an exponential function
The exponent or power x to which the number a is raised to give the number b is called the logarithm of b
to the base a, and be written as loga b z x ax = b
SAMPLES:
1. g 2 +4 3 2 = 9
2. logz 16 24 = 16
Properties of logarithm
1. logx ab = logx a + logx b
1
2. loga b — — a
logb
a
3. logx logx a - logx b
b
4. loge a = In a
5. logx 1 = 0 logxan = nlogx a
In e = 1
SAMPLES:
1. = log* + loga 4
42
2. logz — = 2 log24 - loga
7

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COMPLEX NUMBERS
In a set of all real numbers, the quadratic equation
X2+I=O
has no solution. So we replace the values of x with i thus giving us new equation

This pattern of powers, signs, I's, and ils is a cycle:

3

1 3 = —i

To simplify a square-root radical whose radicand is a negative number:

a) Express the radical as a product of a real and i .
b) Simplify the product using the properties of the real roots of real number
SAMPLE

= -NTT
For all
real numbers a 1.
2. (a + bi)(c + di) = (ac — bd) + (ad + bc)i
3. (a + bi)(a — bi) = a 2 + b 2

ALGEBRA TEST A
Polynomial/Algebraic Expression
1. Evaluate the polynomial 7b 3 + 7b 2 + 6b + 2 if b = 3
(A) 269 (C) 279
(B) 125 (D) 272
2. equal to
(C)2X2 + 2
(B) 2x 2X2
-2

3. Gawing simple ang expression

402
4, — 3) + (8J2 — 2) + (2j4 +j2)
(A) 8j4 + 7j2 + 1
(B)14j4 + 9]
(C) 8j4 + 9j2 + 7
(D) 8j4 + 9j2 _ 5
5. The lengths of two sides of a triangle are given by the expressions 2x 2 + 5x + 4 and 4x 2 + 3.The perimeter
of the triangle is 10x 2 + 3. Find a polynomial expression that represents the lengths of the missing side.
2
(B)4x -5x +2

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(C)4x 2 + 5x + 10
(D)none of these
6. Which of the following is the remainder ofx3 — 3x 2 + 2x — 1 divided by x
(C) -61
(D)61
Special Products and Factoring
7. Which of the following is a term in the expanded form of (x + y) 6 ?
(A) 15x3y3 (C) 5x 3y3
(B) 10x 3 y 3 (D) 20x 3 y 3
8. 313 — 6x 2 — 9x when factored is
(A) 3x(x 2
— 2x — 3) (B) x(3x + l)(x — 3)
2
(C) 3(x + — 3)
(D) 3X(X + — 3)

9.Find x if where a b.
(D) ab
10. Which of the following is not an identity?
(A) (x — + xy + x 2 ) = x3 — y 3
2 2
(B) (ax + by)(cx + dy) = acx + (ad + bc)xy + bdy

(D) (a + b)c = bc + ac
IL Arvin receives 4x 2 — 64 (in pesos) as a stipend from a scholarship every month. If he equally distributes this
amount to food, transportation, photocopying expenses and uniform how much would each expense?

(D) +16
Rational Expression and Equations
12. The multiplicative inverse of b is a, What is the equal toa + b?

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 (c) 2.2
13.
x2+X—42 X2 —36
-2x-13

14. When a camera is in focus, it satisfies the equation — +

(A) 3x2
Radicals
15.Multiply :
(A) SFS — 15
(B) 25€S-
16. Alin sa sumusunod ang hindi rasyonal na numero?

17. Rationalize the denominator:

24+.vfi
(A)
6
16
3
12+
(C) 3
(D)
36—3
a
1
= Find the LCD if p —x , q = x andf=3x2
(C) 6x 3
(D) pqf

(D) 25vfi-1S
= x
(D) no solution
Linear Equation & Linear Systems
1 Y
18. If x= —and 4x = — + 6 then find the value of y.
2 x

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1
2
19. Solve for A
78+61 7B+63
(A)
7
78+18
(B)
2
20. For the what value of a will the linear system have no solution?
2
x—a y=3x—
(a + 2)y = 2
(D)-2
21. Determine the value of 2xy, if x + y = 4 and x2 + y
(D) 16
Quadratic Equations
22. Find the roots of 3x2 + 1 = 6x 3+20 —3—2vfi and
3 3
3+2-vfi 3-2vfi and
3 3

(C)
(D)
23. Solve: 3x2 + 8x = 16

Problem Solving
24. The length of a rectangle is 9 centimeters less than three times its width. If the area is 30 square
centimeters, find the length and width
(A) I = 6cm,w = 6cm
— 7cm, w = 5cm
— 6cm, w = 5cm
(D) 5cm , 2 = 6cm
ag
26. A plane flies 300 miles with a tail wind in 1 hour. It takes the same plane 2 hours to fly the 300
miles when flying against the wind. What is the plane's speed in still air?
(A) 300 mph(C) 150 mph
(B) 225 mph (D) 75 mph
27. How many gallons of a 80% salt solution must be mixed with 40 gallons of a 23% solution to
obtain a solution a solution that is 70% salt?
(A) 92 gallons (C) 18.8 gallons
(B) 9.2 gallons (D) 188 gallons

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28. Tom is 5 times as old as Todd. In 3 years, Tom's age will be 24 years more than 2 times as old as Todd.
How old is Tom?
(A) 45 (c) 42
(B) 43 (D) 46
29. The difference of two number is 28 and their ratio is 3:2. What is the larger number?
(A) 84 (c) 48
(B) 56 (D) 92
Inequalities
30, Ang inequality na —5 x ay maisusulat sa interval na notayon na:

(C) (—00, —5]

31. Graph: 5x— 2

-4 o 1 4

(B)
-4 o 1 4

(C)
-4 o1 4

(D)
32. Which inequality matches the sentence: When a number is increased by 5, the result is less than -2.

Relations & Functions

33. Determine the range of the relation:
{(0,4), (—2, (-31))
(A)
(B) (0.-2,-3}
(c)
(D)
34- Determine which of the following relations is a function:

(B) 16x 2 = 19y2 + 144

(D) 4x2 + = 16
35. Given the function f = (x) = x 2, what is the value of the expression f(f(f(2)))?
(A)128 (c) 32
(B) 16 (D)256
Exponential and Logarithmic Functions
36. Solve: logs(3x + 10) = 4
10 3
(B) 205 (D) 338
37. Kung ang 2 Xay katumbas ng y, ano ang katumbas ng 8X¯ l ?

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PTS Tutorial and Review Systems Katipunan Branch 201a

3
Complex Numbers
38. Express in the form bi where b is a real number.
(D) - 17i

ALGEBRA TEST B
Polynomials/Algebraic Expressions
1. Subtract: (—2d 4 + 4d 3 + 7d 2 ) — (5d4 — 3d2 + d)
(A) —7d4 + 3d3 + 4d2 — d
(B) •—7 d4 3d 3 —d
(C) -7d4 + 5d 3 + -d (D) —7d4 + + 4d2
2. Find f(2) given f (x) = x 3 + 3x 2 — 22
(c) 54
(D) -18
18r—12y3z20
3.

z2S
3y3z2S
6y3z2S 6z25
(B) (D)FF
4. An operation " * " is defined as follows: db = b a . Find the value of (3*2Y2.
(A) 64 (C) 128
(B) 256 (D) 81
5. If x2 + 4 represents an even number what is the sum of the next two consecutive even numbers?
(A)x2 + 4 (A) rectangle
2
(B) 2(x + 7) (B) square
6. Expand: (x + (C) triangle
3
(A) x + 125 (D) impossible to determine
(B) x + 30X + 75X + 125
3 2

(C) + 15 (D) 2X 2 + 10
(D) x 3 + 15x 2 + 75x+125 20x -I- 25, What is an appropriate shape of this paper?
7. Factor: —6x + 24x
4 2

(A) —6x 2 (x + 2)
(B) —6x2 (x2 — 4)
(C) —6x3 (x 2 — 4
(D) —6x 2 (x + — 2)
8. The reciprocal of x — 2 is x + 2. Find x.
9. A piece Of paper occupies an area of 4x 2

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10. Find the volume of a die whose side is (x — l)cm
(A) (x3 — 1)cm3
(B) (x3 + 3x2 + 3r — 1)cm3
(C) (x 3 — 3x 2 + 3x — I)cm 3
3
— x 2 + x — I)cm3
IL Isa lang sa mga sumusunod na rational ekspresyon ang hindi katumbas ng — Ain sa mga ito?

(B) -22
Sn2
12.Find x: is an-s 9n2 —25
(A) 5112 + 311 5
(B)15n3 + 25722
(C) 5n 2
(D) 15112 - 25112
111 1 1
13.If a, b, and c satisfy the equation — = —,then what is c when a — - and b
ab c 3 2
Radicals
14.Simplify: was
(A)27fi
(B) 12Mä6
(c) 15
(D) 42vfi- 15
15. Rationalize:

16. Simplify: VSÖ+VÜ

(C) 2V5 (B) 54Vä
17.
(A)3 16
(B) 36 (D) no solution
Linear, Equations and System
3 3
18. Solve: —3x + — + 5x —
11 11 11
2
11
10
22 11

19. Determine the value of x so that y= — is zero.

(D) y cannot be zero

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20. Solve: 3x + 4y = 10
3x + 4y = 6
(A) (-2,1)
(B) Dependent (many solutions)
(C) (2>1)
(D) Inconsistent (no solution)
21. Find the solution to the system:
(A) (1,1)
(B) (0,0) (D) (2,-2)
Quadratic Equations
22. Solve by completing the square:
x2 — 2x — 24 = 0
(A) {-2,-24) (C) {4,-6}
(B) {-4,6} (D) {2,24}
23. Solve: 3x2 = 25
3 3

24. Determine the nature of the roots.• 3x2 + 30x + 75 = o

(A) two distinct real solution
(B) no real solutions
(C) a unique real solution
(D) cannot be determined
Problem Solving
25. The width of a rectangle is 6 inches less than its length. If the area
of the rectangle is 280 square inches, what is its width?
(A) 36 in. (C) 12 in.
(B) 14 in. (D) 20 in.
26. Janet can paint a kitchen in 4 hours and James can paint the same
kitchen in 3 hours. How long would it take for both working together to
paint the kitchen?
(A) 2— hr (C) 7 hr
12
2 2
(B) 3 hr (D) 1 hr
27. Mia's average driving speed is 8 miles per hour faster than Grace's. In
the same length of time it takes Mia to drive 168 miles, Grace only drives 144
miles. What is Mia's average speed?
(A) 56 mph (C) 40 mph
(B) 64 mph (D) 48
P a g e
28. Ang edad ni Josef ay ikaanim na bahagi ng edad ni Suzy. Pagkaraan ng labing-limang taon, si Josef ay labing
isang taon ang higit sa ikalimang bahagi ng edad ni Suzy. llan taong gulang na ngayon Sina Josef at Suzy?
(A) Si Josef ay 30 at si Suzy ay 5. (B) Si Josef ay 4 at si Suzy ay 29.
(C) Si Josef ay 6 at si Suzy ay
31. (D) Si Josef ay 5 at si Suzy
30.
29. The sum of the digits of a two digit number is seven. If the digits are reversed, the new number is
nine more than the original number. Find the original number.
(A) 25 (C) 43
(B) 34 (D) 16

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Inequality
30. Graph:x +4 S 6 and — 4x < 16

(A)

(B)

(C)

(D)
31. Which inequality does the sentence represent? When a number is divided by-3, and the quotient is
decreased by 6, the result is more than -6.

—3 —6
32. If x is less than 3, which Of the following statements are always true?
l. x is non negative
ll. x is non positive
Ill. 3X is greater than or equal to x
IV. x is less than or equal to x2
(A) Ill only (C) I and Ill only
(B) IV only (D) Ill and IV only
Relations and Functions
33- Which of the following relations is a function?
(A) teacher student
(B) student family name
(C) mother —+ child
(D) child —+ parent
34. Given f(x) = x3 and g(x) = 6 — 5x2 ,find (fog) (x).
(A) (6 - 5x 2) 3 (c)6x3 - 5x5
3
(B) 6 — 5x6
6 —SX2

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35.
Determine which of the following are one-to-one functions.

0
iii iv 0
(A) i, ii and iv only
(B) i and iv only
(C) iv only
(D) iiarzd iv only
69 1
Exponential and Logarithmic Function
36. What is 8¯X if = 81 7
(B) 64
37. The value of x in the equation x = 1 is
Complex Number
38. Perform the indicated operation and write the result in standard form: (3 + 2i) 2
(C) 5 + 12i
(D) —5 - 12i

PLANE GEOMETRY
BASIC TERMS OF GEOMETRY
The word geometry is derived from the Greek word geos meaning earth and metron meaning measure. The basic terms of
geometry are point, line and plane.
(A)point A point represented by a dot and designated by a capital letter next to the dot. •P
(B) Line A line is designed by capital fetters of any two points or by a smaii letter, thus

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(C) Plane A plane is a surface such that a straight line connecting any two points lie entirely on it. A plane is a flat surface and may
be represented by the surface of a flat mirror or top of the desk

KINDS OF ANGLES

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POLYGONS IN GENERAL
A polygon is a closed figure in a plane (flat surface) bounded by straight line segments, called sides.
Name of Polygons According to the Number of Sides

No . of Sides Polygon No. of Sides Polygon

3 Triangle 8 Octagon
4 Quadrilateral 9 Nonagon
5 Pentagon 10 Decagon
6 Hexagon 12 Dodecagon
7 Heptagon n-gon
An equilateral polygon is a polygon having equal sides. Thus, a square is an
equilateral polygon.
An equiangular polygon is a polygon having equal angles. Thus, a rectangle is an
equiangular polygons Regular Polygon
A regular polygon is an equilateral and equiangular polygon. Thus, a regular pentagon is a 5-sided equilateral and
equiangular polygon.
D

Octagon
Sides) 0
135X8=1080

CIRCLES
A circle is the set of points in a plane at a fixed distance, called the radius, from a fixed point, called the
center.
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The circumference of a circle is the length of the circle. Circumti:rence Segment

An arc is a curved line that is part of a circle.
A radius is a line segment joining the center of a circle to a point on
a circle. All radii of a circle are equal.
A chord is a line segment joining any two points of a circle
A diameter is a chord through the center of a circle. A diameter is the
longest chord. It is twice the length of a radius.
A sector is a figure bounded by an arc of a circle and the two radii to
the ends of the arc.
Circles, Arcs, Tangents and Secants
1.The measure of an intercepted arc is equal to the measure of Diameter
the central angles

n AOB
2.An inscribed angle js equal to half the intercepted arc.
3 The measure of an angle formed by 2 chords is equal half the sum of intercepted arcs
LRAK = + AR)
4. An angle outside a circle formed by either a secant and a tangent, 2 tangents and secants is
equal to half the difference of the intercepted arcs,
LEAN = - ( fN A)

TRIANGLES
An equilateral triangle has three equal sides. It also has three equal angles, each 60 0
An isosceles triangle has at least two equal sides. It also has at least two equal angles.
The equal angles shown lie along the base b and are called base angles A scalene
triangle has no equal sides.
A right triangle has one right angle. Its hypotenuse is opposite the right angle. Its legs (or arms)
are the two sides. The symbol for the right angle is a square corner.
An obtuse triangle has one obtuse angle (more than 90 0 and less than 180 0) An
acute triangle has all acute angles (less than 900).

Isosceles Triangle Scalene Triangle Equilateral Triangle Right Triangle

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Acute Triangle Obtuse Triangle

UNDERSTANDING QUADRILATERALS
Quadrilateral just means "four sides" (quad means four, lateral means side).
Any four-sided shape is a Quadrilateral.
But the sides have to be straight, and it has to be 2-dimensional.
Properties
Four sides (or edges)
Four vertices (or corners).
The interior angles add up to 360 degrees:Types of Quadrilaterals
There are special types of quadrilateral:

Trapezoid (US)
Parallelogram Rectangle Rhombus Square Trapezium (UK) Kite
Some types are also included in the definition of other types! For example
a square, rhombus and rectangleare also parallelograms.
The Rectangle
means "right angle"
I and I l show equal sides
A rectangle is a four-sided shape where every angle is a right angle (900 ).
Also opposite sides are parallel and of equal length.
The Rhombus

A rhombus is a four-sided shape where all sides have equal length.

Also opposite sides are parallel and opposite angles are equal.
Another interesting thing is that the diagonals (dashed lines in second figure) of a rhombus bisect each other at right
angles.

The Square
means 'Fright angle"
show equal sides
A square has equal sides and every angle is a right angle (900)
Also opposite sides are parallel.

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A square also fits the definition of a rectangle (all angles are 900 ), and a rhombus (all sides are equal length).
The Parallelogram

A parallelogram*s opposite sides are parallel and equal in length. Also opposite angles are equal (angles "a" are the
same, and angles "b" are the same)
NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!
The Trapezoid

Trapezoid Isosceles Trapezoid

A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.
It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a
parallel side are equal.

The Kite

Hey, it looks like a kite. It has two pairs of sides. Each pair is made up of adjacent sides that are equal jn length. The
angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal
bisects (cuts equally in half) the other.
Irregular Quadrilaterals
The only regular quadrilateral is a square. So all other quadrilaterals are irregular.
The "Family Tree" Chart
Quadrilateral definitions are inclusive.
Example: A square is also a rectangleSo we include a square in the definition of a rectangle. (We don't say
rectangle has al/ 900 angles, except if it is a square")
This may seem odd because in daily life we think of a square as not being a rectangle ... but in
mathematics it is.
Is a Square a type of Rectangle? (Yes)
Is a Rectangle a type of Kite? (No)

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Rhombus Rectangle

Square
Perimeters
Polygon perimeter
Equilateral Triangle
Isosceles Triangle
Scalene Triangle a+b+c
Square
Rectangle 2b+2h
Obtuse Triangle a+b+c
Regular Pentagon
Regular Hexagon 6s
Regular Heptagon
f

AREA AND VOLUME

FORMULA SHAPES FORMULA
Parallelogram a = base x height Cube 3
S = 6S 2

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Rectangle Sphere 4
a = lw
p = 21 + 2w 3
S = 47tr 2

Triangle bh Cylinder V = Ttr 2 h

2 s = 2grh

sum of all angles 1800

h

Equilateral Triangle Rectangular Prism v = lwh

4
p — 3s s = 21w + 21h + 2wh

s h

Trapezoid Cones
a
2 1
V = —Ttr 2 h
c h 3
S = Ttr2 +

Circle c = dn = 2,r7t Pyramid

2 3
s
= base area
+ lateral surface area
2
Rhombus 2 Square

s s

ANALYTIC GEOMETRY
RECTANGULAR COORDINATE SYSTEM
Coordinate Plane is a plane determined by the coordinate axes. Coordinate axes
consist of x and y axis. The coordinate axes divide the plane into four parts, called
quadrants namely l, Il, Ill and IV.
Quad I y-axis
x-axis Quad Il rigin

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A point is represented with coordinates x and y or ordered pairs (x,y), where x is the x-value or abscissa, and y is the
y value or ordinate.
LINES AND FIRST-DEGREE EQUATIONS
The equation of every straight line is expressible in terms of the first degree. The graph of a first degree equation is
a straight line.
Let (xl,y,) and (x2,Y2) be two points of a line. The slope of a line is denoted by m, and is equal to the
formula.
Y2-Y1 . rise X2-Xl run
SOLVE:
Find the slope of the following given two points
1. (3,0) and (2, -3)
2. (1.4) and (-5,6)
3. (0,4) and (0,-1)
4. (5,0) and (-1,0)
Line Intercept
1.x-intercept is the x value, when
2.y-intercept is the y value, when
Equation of a line
Slope Intercept Form y = mx + b
Where m m is the slope and b is the y - intercept
Sample:
Find the equation of the line given the following:
1. m 2 and y intercept is -1
Solution: using slope-intercept form Y = mx + b

2x —y = 1 (standard form)
2. m = —3/4 and y intercept is 2 SOLUTION:

3x + 4y = 8 (standard form)
ll. Point-Slope form: y = m(x Xl)
Where (Xl, Yl) is a point and m is the slope.
Sample:
Find the equation of a line given a point and slope.
I- A(4, —2)and m = —2
2
SOLUTION: Y -Yl -
1
2
2y + 4 — — 2y +
4 — —x + 4 x + 2y
2. B(—I,O) and m = —4
— —
m(x — Xl)

SOLUTION:

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Y2-Y1
Two-Point Form y —
Sample:
Find the equation of the line determined by the following points
1. (3, —3) and (2,4)

SOLUTION: —y 21
7x+y = 18 (standard form)
1
2. (3,0) and (4,0)
(0—0) SOLUTION:

y = 0/1(x- 3) y = 0
(standard form)
Distance Formula
To find the length or the distance between two points (XI.YI) and (x2, h), we use the formula

Midpoint Formu la
To find the value of the order (E, ')ofthe midpoint M of a line segment where (x2,Y2) are the endpoint,
we use the formula
Xl -f- X2 Yl+Y2
(Eli) = ( 2 2)
CONICS
Conic Sections are the curves, which can be derived from taking slices of a "double-napped" cone
CIRCLE
A circle is a set of all points in a plane that are equidistant from a fixed point on a plane. The fixed point is
called the center, and the distance from the center to any point of the circle is called radius. The standard
form of a circle is
— r where (h, k) is the center and r is the radius
SAMPLE:
1. Find the equation of the circle of radius 4 which is centered at (3,-2).
SOLUTION:
If the center if a circle is a (3,-2) and radius is 4, the equation of the circle is (x
— + (y - = 42 x 2 + Y2 — 6x + 4y — 3

The last equation is of the form

This is called the general form of the equation of the circle.

2. Find the center and radius of the circle with general equation
2x 2 + 2y 2 — 8x + 5y-80 = o
SOLUTION:
5
25

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=
40
+4
+

16
go
PARABOLA
A parabola is a set of all points in a plane that are equidistant from a fixed point and a fixed line of the plane, The
fixed point is called the focus and the fixed line is the directrix.
The equation of a parabola with vertex at the origin and focus at (a, 0) is y 2 = 4ax
The parabola opens to the right if a > 0 and opens to the left if a < 0.
The equation of a parabola vertex at the origin and focus at (Ora) is x 2 = 4ay The
parabola opens upward if a > 0 and opens downward if a < 0.
Parabola with vertex at (h,k)
The equation of a parabola with vertex at (h, k) and focus (h, k + a) is (y —
= 4a(x — h)
The parabola opens to the right if a > O and opens to the left if a < O.
The equation of a parabola with vertex at (h, k) and focus (h, k + a) is (x —
= 4a(y — k)
The parabola opens upward if and opens downward if a < 0.
SAMPLES:
1. Write the equation of the parabola with vertex at the origin and the focus at (0,4).
SOLUTION: The distance from the vertex to the focus is 4, and hence a=4. Substituting the value for a, we get x 2 =
16y
2. A parabola has its vertex at the origin, its axis along the x-axis, and passes through the points (-
3,6). Find its equation.
SOLUTION: The equation of the parabola of the form Y 2 = 4ax. To determine the value of 4a, we substitute the
coordinate of the given point in this equation. Thus we obtain 36 = 4a(—3) and 4a = —12
The required equation IS y
ELLIPSE
An ellipse is a set of all point P in a plane such that sum of the distance of P form two fixed point F l and F on the
plane is constant,
The standard equation of an ellipse with center at the origin
+
= 1 major axis is the x axis
x
z + • = 1 major axis is the y axis
If the center is at point (h, k), the standard equation of an ellipse is a2 b2 = 1 when major
axis is parallel to x axis.
Similarly, when the major axis is parallel to y-axis, we have a2 SAMPLE:
Reduce the equation to standard form
4y 2 + 9x2 - 24y— 72x + 114 = O
SOLUTION:
402 — 6y) + 9(x2 — 81) — —114
402 — 6y + 9) + 9(x2 — + —114 + 36 + 144
9 4 36

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HYPERBOLA
A hyperbola is a set of points in a plane such that the difference of the distances of each point of the set from two
fixed points (foci) in the plane is constant.
The generalized equations of hyperbolas with axes parallel to the coordinate axes and centers at (h,k) are:
SAMPLE:
Find the center of the hyperbola whose equation is 12y 2 — 4x 2 + 72y + 16x + 44 = 0 SOLUTION:
1202 + 6y + 9) — 4(x2 — 4x + 4) = —44 + 108 16
12(y + 48
(Y +
3)2
4 12
Center is at (2, —3)
GEOMETRY TEST A
Basic Terms of Geometry
I. If two lines coincide, then their intersection contains
a) only one point
b) two points
c) infinitely many points
d) another line
2. In a triangle ABC, AB BC. P is any point between A and B. which of the following is always true?
a) Ipcl
b) IAPI > IPCI
c) 'PCI > IAPI
d) IAPI Ipcl
3. Alin sa sumsunod ang tama
a) mal + mL2 = 900
b) = 360 0
d) mal + mul = 180 0
4. The difference between the measure of an angle and its supplement is fifty-two. Find the measure of both angles.
a) 1280 and 52 0
b) 1160 and 640
c) 136 0 and 840
d) 71 0 and 190
5. Given: if mL1 = 6x - 58 mL4 —4x— 22 then find the value of
a) 24
b) 25
c) 26
d) 27
6. If the measure of one angle of an isosceles triangle is 200, then no
angle of this triangle can have a degree measure of
a) 80 0
b) 20 0
c) 40 0
d) 1400
7. Alin sa mga sumusunod ang hindi maaring sukat ng mga gitid ng isang right triangle?

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a) 3,4,5
c)
d)
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8. The measures of the sides of a triangle are 10, 13 and 13 cm. How far is the shortest side to the
opposite vertex?
a) 11 cm c) 12 cm
b) 12.5 cm d) 13 cm
9. What is the diagonal of a square with side 4cm?
d) 16
10. The side of a right triangle are x, x + 17, and x + 18 unit long, What is the length of the hypotenuse?
a) 24 units
b) 7 units
c) 25 units
d) 11 units
11. One leg of a right triangle is 4 inches longer than the other leg. The hypotenuse is 20 inches long. How long
is the shortest leg?
a) 14
b) 16
c) 11
d) 12
12. Ang bilang ng "diagonal" ng isang hexagon ay?
13. If mL4 = 47 and ml-5 = x + 110 find the value of x.
a) 17
b) 25
c) 23
d) 15
14. The perimeter of regular heptagon with is
a) 6x
c) 14x
d) 12x
Problem Solving
15. The length of a rectangle is three more than twice its width. What are the dimensions of a rectangle if its
perimeter is 138 cm.
a) 22 cm and 44 cm
b) 29 cm and 31 cm
c) 22 cm and 47 cm

21 cm and
46 cm

h
a 16. Given an
isosceles trapezoid, if Ä = 10cm, = 14cm and n = VS then the perimeter of an isosceles trapezoid is:

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a) 27 cm
b) 30 cm
c) 28 cm
d) 34 cm b
17. What is the length of each edge of a cube if the sum of all its edges is 240 cm?
a) 10 cm
b) 20 cm
c) 8 cm
d) 5 cm
18. If a circle of diameter 8 has the same area as a triangle of a base 4, what is the altitude of the triangle?
a) 47t
b) 6Tt
C) 8Tt
d) 10T
19. The length of a rectangle is twice the width and is equal to the side of a square. What is the area of the
square?
a) 8w 2
c) 4w2
20. If the lengths of two sides of a right triangle are 3cm and 4cm, what is the length of the third side?

21. The area of the circle with radius r is ITT 2 . If the radius of the circle is tripled, then what would be the ratio
of the area of the old circle to the new circle?
1
9
1
6
c) 3
2
3
22. If a square has a perimeter of 32 inches, then what is the area of the inscribed
circle?
a) 8m sq. inches
b) 16T sq. inches
c) 4T sq. inches
d) 20m sq inches
J
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23. Kung ang bilog ay nakapaloob sa isang parisukat na may gilid na 7, ano ang sukat-
laki ng bahaging walang shade?
a) 49 — 49T yunit kuwadrado
b) 49 — 12 —Tt yunit kuwadrado
c) 12—1T — 49 yunit kuwadrado
d) 49 + 49T yunit kuwadrado

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24. A cylindrical tube has a volume of161rcm 3 and a cross-sectional area of 2m:m 2 . Find the height of
a cylindrical tube.
d) 12
25. The side of a cube measure 3 cm. Find the surface area.
c) 48cm2 c) 72cm2
d) 54cm2 d) 98cm2
26. In then figure on the right, AD 4,DB = 6 and ¯ DE = 8, the value of x is
BC//DE
a) 10 c) 20
d) 15

c
27. Two similar triangles have areas 2k 2 and 8k 2 . What is the ratio of corresponding sides?
c)1.4
28. A circle centered at Q with a radius of 7 overlaps a square as shown in the
picture. What is the area of the shaded region? Use = —
a) 10.S c) 38.5
b) 27 d) 49
Analytic
Geometry
29. Which quadrant
or axis contains
the ordered pair (0,-
31. Which 4)?
a) Origin
b) X-axis
c) Y-axis
d) Quadrant Ill
30. Name the quadrant or axis containing
the point (-1,4).
a) Quadrant lil
b) Quadrant Il
c) Quadrant I
d) Quadrant IVa)
b)
c)
d)
32. In the triangle given,
AD is an angle bisector.
LDAC is 300 and LABC is a
right angle. Find the
measure of angle x.
a) 300
b) 450 c) 1200
B

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33. Determine the midpoint of a line segment whose endpoints are (9,-2) and (3, 4)
a) (6,1)
c) (3,-3)
d) (11,-1)
34. What is the equation of a line in standard form with slope O and y intercept -1.

c) x + 1 = 0
9
35. What is the equation of a line, in slope-intercept form, with slope and y-intercept 4. 5

9
9 c) y = —-x—4
5

36. Graph 3y + 12 = O
a)
b)

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37. What is
the slope
of a

39. Which

horizontal line?
c) Undefined
d) Cannot be determined
38. Give the slope of the line that contains (-8,4) and (-8,-8).
b) Undefined
3 c) 4
a)
b)
40. What is the slope of the line 9x— 3y — 547
a) 3
b) -3
c) -6
d) -18
41. Find the x-and y-intercept of y
a) x-intercept -8; y-intercept: 8
b) x-intercept -8; y-intercept.
c) x-intercept 1; y-intercept: -8
d) x-intercept 8; y-intercept: -8
42. The graph of the equation Y2 + x2 = 4 is
a) Line d)
b) Circle
c) Parabola
d) Ellipse
43. Alin sa sumusunod ang hindi sekyon ng "conic"?
a) Circle
b) Parabola
c) Ellipse

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d) Cone
44. What is the center of the circle in the equation (x
a) (2.1)
b) (2,-1)
c) (-2,1)
d) (-2,-1)
45. What is the radius of the circle passing through points (7,-1). (It-I), and (4,2)?

46. The parabola with equation Y 2 = x — 4 has a graph opens

a) To the right
b) To the left
c) Upward
d) Downward
47. Which of the following is an equation of an ellipse?
a) x2 — = 4
b) x 2 + 2 = 1
2 c)

4 2

48. The center of the hyperbola with equation 9 4

a) Origin
b) (3,2)
c) (9,4)
d) (1.1)
1
49- Find the roots ofthe equation: x
a) O and 3
b) 3 and -3
c) and —
d) O and -3

SO. Which of the following is true regarding the two lines whose equations are:
a) The lines are parallel
b) The lines are perpendicular
c) The lines intersect at a point on the y-axis
d) The lines coincide

GEOMETRY TEST B
Basic Terms of Geometry
1. Given 5 points A, BJ C, DJ E how many line segments can be drawn by joining exactly two points, that is, no
three points are collinear?
a. 8
b. 10
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c.
12
d. 15
2. The base angle of an isosceles triangle has a measure of x o. What is the degree measure of the vertex
angle?
a. 180 — 2x 90 —
c. 90 -2x
d. 180 —E
2
3. Sa tatsulok na nasa larawan, ano ang kabuuang sukat ng tatlong
exterior angles na may marka?
a. 7200
b. 6000
c. 9000
d. 10800
4. If the minute-hand of the clock coincides with the hourhand, then
what is largest angle formed between the two hands of the clock?
a. 00
b. 90 0
c. 1800
d. 3600
5. What is value of x if AB, CD, EF are concurrent? mLEOC = 5x - 18 nuCOB —— 2x + 9 mLAOE = 2 + lox

a.
b. 10
c. 11
d. 12
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6. In the diagram below, If AB > CD which of the following must be true?

AB > CD
ll.
Ill. AC > CD
a. I only
b. i and li onfy
c. Ill only d l, Il and Ill
7. Which of the following is not true about any right triangle?
a. The sum of interior angles is 1800
b. The area is half of the product of the height and base.
c. The hypotenuse is the squareroot of the sum of the square of the legs.
d. The measure of one angle is more than 90 0
8. If the sum of the measure of two angles of a triangle is half the third angle, what is the measure
of the third angæe?
300 450
c. 600
d. 1200
9. The ratio of the angles of a triangle is 2:3:4. The smallest angle in the triangle is:
a. 250 b. 300 400 d. 600
10. The measure of the exterior angle at the base an isosceles triangle is 130 0. What is the measure
of the vertex angle?
a. 500
b. 600
c. 700
d. 800
11. Ang tuldok na S ay nasa pagitan ng R at T. Ang sukat ng
"angle" URS ay 2x + I, ang "angle USV ay 3x at ang "angle" VST
ay 4x — 1.
Ano ang halaga ng x kung ang tatsulok ng RUS ay isosceles?
300 b. 20 0
c. ISO
d. 40 0 s
12. The sides of a right triangle are x, x + 3 and x + 6 units
long. Find the side lengths of the triangle by solving x.
a. units
b. 9,3,15 units
c. 3,15,6 units
d. units
13. The sides of a right triangle are x, x + 7 and x + 8 units long. What is the length of the hypotenuse.
a. 5 units
b. 13 units
c. 3 units
d. 12 units
14. Ang pentagon ay may ilang bilang na diagonal?
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4
b.
c. 6
7
15. Ang parihaba na may haba na apat na sentimetro ay may lawak na katulad ng sa parisukat ng
may gilid na walong sentimetro. Ano ang perimeter ng parihaba?
a. 40 sentimentro
b. 35 sentimentro
c. 30 sentimentro
d. 20 sentimentro
16. The width Of the rectangle is half the length and is equal to the side a square. The perimeter of a square is:
a.
b. 2L
c. 4L
17. The perimeter of a rectangle is 70mm. the height is one less than three times the base. What is the length
of the base?
71
a.
4
b. 23 9
d. 32
18. Find the area an isosceles trapezoid in the figure on the right if
BC = 8cm, AE = lcm, BA = 2cm.
a. 9Nficm 2
b. 18Gcm2
c. 9 cm2 D
d. 18cm2
19. The ratio of the side of two squares is 1:2- If the area of the smaller square is 16, what is the area
of the larger square?
a. 49
b. 64
c. 91
d. 100
92 1!
20. A square with length 4 is circumscribed in a circle. What is the circumference of the circle?
a. 411
b. 16T
c.
d. 27T
21. 17m
How long is a wire 14mfrom
stretched
—
+ 5 the top of a fence (4m-2) meters high to a point on a ground (m+l)

meters from its base? 2

22. Find the area of shaded region, if a rectangle has dimensions 8 and 12 and the circle is inscribed in the
rectangle.

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a. 12(8 371)
b. 12(8 + 371)
c. 16(6 — it)
d. 16(6 + TV)
23. Find the perimeter of the figure below. (sq-square)
a. 5a+b+c
b. 17a+b+c C. 13a+b+c
12 a +b+C
24. Batay sa larawan, ang dalawang bilog ay concentric. Kung
ang diameter ng maliit na bilog ay 12 at ang sukat mula sa dulo ng
diameter at ng malaking bilog ay 2, ano ang sukat laki ng may "shaded"na
bahagi?
a. 127T
b. 28m
c. 10T
d. 16T
25. A larger cube has an edge of 8cm. If the ratio of the surface areas of the two cubes is 1.•2, What
is the total surface area of the smaller cube?
a. 384ctn2
b. 192cm2
c. 6 cm2
d. 32cm2

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26- What is the surface area of a rectangular prism shown below?

a. 200cm2
b. 400cm2 10 cm
c. 110cm2
220cm2
27. What is the length of rc? D Ac
mLC = mLFl- 5 em •l cm
mLA = mLD
AB : DE = 5 : 11
125 b. 605
c. 315
d. 420
28. The length of the sides of XYZ are 74, 50 and 48. The length of the smallest side of XAB is 216, what is
the length of the longest side ofXAB?
mLY = 171LA
a. 16.52
b. 333
c. 222
d. 123
Analytic Geometry
29. Name the quadrant or axis containing the point (-8,-8)
a. Quadrant Ill
b. Quadrant IV
c. X-axis
d. Quadrant I
30. What are the coordinates of the point shown in the figure on
the right?
a. (-5,2)
b.
c. (2,5)
d. (2,-5)
31. Use the table to choose the correct linear equation;
x

a.
d.
/

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32. If the coordinates of two opposite vertices of a rectangle are (3,3) and (7,9), which of the following are the
possible coordinates of the other two vertices?
a. (7,3) and (3B)
b. (2,8) and (7,3)
c. (2,8) and ( 8,4)
d. (3,9) and (8,4)

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33. What is distance between A and B if A and B have coordinates (-4,-6) and (2,-1) respectively?
c. 61
d. 11
34. — 3x s-- 8y = —4 expressed in slope-intercept form is
b.
4
c.
8 3
3 1

8 2
35. What is the standard form of the line with slope -3 and y-intercept -4.

b.

d.
36. The vertex of x = — is located at
b.

c.
37. The endpoints on an ellipse defined by equation 4(x— I)2 25 (y 2) 2 —— 100 are
b.
c.
d.
38. The maximum value of y for the equation y 6
c. 3
d.
39. Which shows the graph x + y = 8.

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 d.
40. What is center of an ellipse with equation 9 2
a. (1,0)
b. (0,-1)
c. (-1,0)
d. (0,1)
41. What is center of the hyperbola with equation 4(y 2 + 6y + 9) - 9x 2 = 36?
a. (9,0)
b. (029)
c. (0,-3)
d. (-3,0)
42. Find the y-intercept of the line 3x —12?
a. -3
b. -4
c. 4
d. 3

43. Graph x =

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44. The slope of the line passing through the points A (-2.8) and BC-8,3).
11
10
12
45. What is the slope of the line y

46. Classify the conic section x

a. hyperbola ellipse
c. circle
d. parabola
47. Which of the following is not an equation of conic?
a. x2 + Y 2 + 8x+ 16y + 16 — 5
2
b. x 2
+y 2 — 8x = 4 C, 4y —l- 15
48. What is a equation a circle with the given 4 and center
a. (x + _ (y + = 16
b. (x + + (y + = 16
d. (x— + (y— = 16
49. Determine the center and radius of a circle with equation x 2
+ Y 2
+ 4x -
6= 0

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 d. c(O,2),
50, What is the standard form of a parabola with equation
2

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TRIGONOMETRY
Trigonometry is a branch of mathematics that deals with the study of the measure of the triangle. It was derived from
the greek words: tri, which means three; gonia, which means angle; and metron, which means to measure,
TRIGONOMETRIC/ CIRCULAR FUNCTIONS
Consider a right triangle, we refer to the three sides as the hypotenuse, the opposite side, and the adjacent side whose
lengths are r, y, and x respectively.

adjacent side x
CIRCULAR FUNCTIONS RECIPROCA
L
opposite side hypotenuse
sine = csc8
hypotenuse opposite side
adjacent side x hypotenuse
cos = sec B =
hypotenuse
opposite side adjacent side x
Y tan 0 = adjacent side x
adjacent coto =
side x opposite side
PYTHAGOREAN THEOREM
c2 = a 2 + b 2
a

b
EXAMPLE:

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Find the value of the remaining functions given that

3 sino =
5

3
b
EXAMPLE:
Find the value of the remaining functions given that sin e = 3 5
SOLUTION:
opposite 3 Since sin 6 =
hypotenuse 5
To get the length of adjacent side, we use the Pythagorean Theorem.
3 2+ b 2= 5 2
b = 4 adjacent side
Therefore,
5
3 si CSC 6 =
n6=— 3
5 5
4 co sec9
se — =
3 4
tan e =
4 4
coto =
3
ALGEBRAIC SIGNS OF CIRCULAR FUNCTIONS

Sin All

cos
an

NOTE:
In quadrant I, All functions are positive
In quadrant Il, Sine and cosecant functions are positives. All other functions are negative
In quadrant Ill, Tangent and cotangent functions are positive. All other functions are negatives
In quadrant IV, Cosine and secant functions are positives. All other functions are negatives.
Reference Angle is the absolute value of the measure of the acute angle from the x-axis
SAMPLE: —300 0 has 600 as a reference angle

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 /
Coterminal Angles are angles which, drawn in standard position, share a terminal side. For example, 600, -
3000, and 420 0 are all coterminal.
For a given angle, a multiple of3600 is added or subtracted to the given angle.
SAMPLE:
200 could have —340% 380 0, 720 0, etc. as the coterminal angle.
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
Reciprocal Identities
sin = cos g
cos 9 =
sec 0
tan 0 = cot e
Quotient Identities
sin e cos tan 8 = cote =

cos9' sin 8
2
Pythagorean Identities sin + cos
2
9=1
1 + tan2 0 = sec2 9
1 + cot2 0 = csc2 0
CIRCULAR FUNCTIONS OF SPECIAL ANGLES

sin

sec

|
ANGLE OF ELEVATION/ANGLE OF DEPRESSION
plane

A. ANGLE OF ELEVATION
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 ground level B.
ANGLE OF DEPRESSION

observer

object
SAMPLE:
From a point 542 feet in a horizontal line from the base of a building, the angle of elevation of its top is 30 0.
What is the height of the building?
SOLUTION:

542ft
tan 30 0 =
542ft
Y = (tan
542vfi

Y = ¯ft
3
CALCULUS
Limit (Informal Definition)
Let's look at the sequence whose n th term is given by n/(n+l).
What will this sequence look like?
1/2, 2/3, 3/4, 4/5, 5/6,... 10/11,... 99/100,... 99999/100000,...
What's happening to the terms of this sequence? Can you think of a number that these terms are getting closer and
closer to? Yep! The terms are getting closer to Il But, will they ever get to 1? Nope! So, we can say that these terms
are approaching 1. Sounds like a limit! The limit is 1.
As n gets bigger and bigger, n/(n+l) gets closer and closer to 1...
We then say that f(n) approaches the value I as n approaches infinity, or that the limit of f(x) as x increases infinitely
is equal to I.
Symbolically we write limits as

You will learn more of this in the college level.

For "well-behaved" functions (e.g. polynomials), we can evaluate limits by direct substitution.

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Example 1. Consider the polynomial function p(x) = (2x 3 — 5x + 1) then by direct substitution we get lim p(x) =
lim (2x3 — 5x + 1) = 2(2) 3 — 5(2) + 1 = 7.
x -2x+1
Example 2. For f (x) — by direct substitution we get
3x+6 x2—2x+I lim f (x) = lim
x—y3 3X+6

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15"

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TRIGONOMETRY TEST A
Degree, Radian, Revolution
11
1. Convert to degrees: —IT
18
550
b. 2200
c. 1100
d. 2950
2. How many revolutions does a second hand of a clock make every 24 hours?
a, 60
b. 3600
1440
d. 86400
3. What is the coterminal angle of 270 0 ?
a. 5400
b. 6300
c. 7200
d. 9000
2

4. If sine = — then what is the value of cos6?

5
a.
b.
2
c.
5
1
d.
5
Algebraic Signs of Circular Functions
For numbers 5 and 6, identify the quadrant or quadrants for the angles satisfying the following conditions:
5. sin a > 0, cosa < 0
a.
c.
d. IV
6. tan 9 > 0
a.
b. Il or Ill
d. t or Ill
Values of Trigonometric Functions
7. Which of the following is not true about angle 9, if cos 9
2 kit
b. 0 is a multiple of—, k is an odd number
2
9 = 2700
37t
8. Evaluate: cos—
2

1
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b.
c. -1
d. Undefined
9. Fill in the blank: sine = a, s in2 6
b.
c. 1
d. tan29
10. Alin sa mga sumusunod ang hindi tugmaan?
1
a. = sec x COS X
b. cosx + sin x 1
c. tan2x + 1 = sec 2 x
sin x d. =
tan x
cos x
Solution of a Right Triangle
11. From a point on the ground 10m away from the Math Building, the angle of elevation from the top of
Math Building is 300. Determine the height of the building.
10
a. —m 3
b. 3 v'fim
c.
IOUS
3
12. Two townhouses are 3m apart. The angle of depression from the roof of the taller townhouse to the roof
of the other townhouse is 60 0. How high is the taller townhouse if the other townhouse is 4m high?
a. 6m
8m
c.
d.
Evaluating Limits
13. Find: + 2x2 + 27)
59
43 c. 55
d. None of these
x3—64
14. Find: lim x4—2S6
1 a.
18
3 b.
8
3
c.
1 d.
4
Derivative
15. Find the derivative of y = 4x3 .
12x
b. 12x2
12x3log I

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TRIGONOMETRY TEST B
Degree, Radian, Revolution
23
1. Convert to degrees: —Tt
12
3450 b.
6900
c. 6800
d. 3350
2. On a clock with hour, minute and second hands, the second hand is 10 cm long. In 15 minutes the tip of
the second hand travels a distance of XTtcm. Find the value of x.
a. 75
b. 150
300 d.
250
19
3. In which quadrant does —TC lie?
c.
d. IV
Trigonometric Functions
4. Given the right triangle below, which of the following represents the value of x?
a. 5sinß0
b. 5 cosß0
5 C. cosßO
d.
sinßO
Algebraic Signs of Trigonometric Functions
5. Which of the following conditions is positive?
cos8, 90 < e < 180
sing, 180 < ß < 270 c.
cota, 180 < a < 270
d. cscy,270 < Y < 360
6. In which quadrant do cos9 < 0 and sine < 0 lie?
a.
b.
c. Ill
d. IV
Values of Trigonometric Functions
7. What is the value of sin —?
6
1
a.
b.
2
1 c.
2
d. 1
8. If sin600 = — and cos600 = what is cot60 0 ?
2 2'
2
b.
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c.
d. 1
Fundamental Trigonometric Identities
9. Fill in the blank: tan2 6 =
a. csc29
b. sec2e
c. sin2B
d. sece
10. Which of the following is not equal to sine function?

b. tanx cosx
c. tan x cos x
1
d. cos x sec x
CSC x

11. A pole casts a shadow 3m long. If the height of the pole is 4m, how long is the rope tied from the top
of the pole to the tip of its shadow?
a. 3m
4m c. 5m
d. 6m
12. Sol places a ladder next to a building to get to the top. The bottom of the ladder is placed 15 feet
away from the building at an angle of 600. How long is the ladder?
a.
b.
IS
c.
2
d. 30
Evaluating Limits
13. Evaluate; limx-.—l (2x2
a. -2 b. 5
4 d. 3
x3—1 14.
7 b. 6 c. 5
-6
Derivative
15. Find the derivative of y = --6x—1
a. 6x
6
b.
x2
6
c.
x2

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WORD PROBLEMS
The general procedure of solving word problems is done by expressing the conditions of the problems in algebraic
symbols. No definite rule can be followed for solving word problems but the following suggestions will help in the
translation of word problems into mathematical symbols.
Strategy for Solving Word Problems
1. Read very carefully the problems until the conditions are all clear. Determine the given and set
what is to be found.
2. Draw figures or diagrams and label the known and the unknown parts.
3. Use formulas connecting the known quantities with the unknown quantities.
4. Represent the unknown quantities in terms of x.
5. Form an equation of the translated condition of the problem by relating the known and the unknown
quantities.
6. Solve for the unknown from the formed equation7. Check the solutions with the condition of the
problem.
NUMBER RELATION PROBLEMS
The sum of two numbers is 36. If the larger number is divided by the smaller number, the quotient is 2 and the
remainder is 3. Find the numbers.
SOLUTION:
Let x = the larger number, and
36 — x = the smaller number
The equation is:
x 3

36 —x 36 -x x = 25
Therefore, X = 25 is the larger number and 11 is the smaller number ANOTHER
SOLUTION:
Let x smaller number
36 — x the larger number
The equation is:

39 — x 3 x
x x = Il
Therefore, x = 11 is the smaller number, and 25 is the larger number.
The sum of two consecutive integers is 15. Find the number.

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INVESTMENT PROBLEMS
Interest is the money paid for the use of money called principal. In general, interest is determined by the
product of three factors — principal, rate and time.
The principal is the sum of money invested that bears interest.
The rate is the fraction of the principal paid for its use during a certain period of time, usually one year.
The time is the interval during which the principal is used.
The amount is the sum of the principal and interest.
FORMULA TO REMEMBER:
Simple Interest I = Prt
Compound Interest

EXAMPLE:
Cheska has P6,000 invested at 5% and 6%. How much would she have to invest at 6% so that her total interest per
year would be equal to P320?
SOLUTION:
Let x = the amount invested at 5%
6000 — x = the amount invested at 6%
The interests for each investments are
= interest at 5% =x(0.05)(1)
= interest at 6% = (6000 —
Since the annual interest is P320, then
= P320 or + (6000 — = 320
Solving for x, we get x = 4000, amount invested at 5% and P2,000, invested at 6%.
A man divides PIO,OOO in two investments, one at 10% and the other at 30%. Find out how much is invested at
each rate so that the two investments produce the same income annually.

P
COIN PROBLEMS
The nature of solving coin problems is related to that of mixture problems. The guiding principle here is the fact
that the sum of the values at each denomination is equal to the total amount of coins.
EXAMPLE:
A coin purse contains 5c, IOC and 25c coins. The number of IOC coins is three times as many as 5c coins,
and the 25c coin is two more than the IOC coin. If the total value of its contents is P4.90, how many of
each kind of coins are in the purse?
SOLUTION:
Let x = the number of 5c coins 0.05x = the value of 5c coins
3x = the number of IOC coins 3x(o.10) = the value of IOC coins
3x -f- 2 = the number Of 25c coins = the value of 25c coins
The equation is based on the value of 5c + the value of IOC + value of 25c = P4.90
0.05x + 3x(o.10) + (3x + 2) (0.25) = 4.90
Solving for x, we get x = 4, for the number of 5c coins, twelve IOC coins and fourteen 25c coins,
Your uncle walks in, jingling the coins in his pocket. He grins at you and tells you that you can have all the
coins if you can figure out how many of each kind of coin he is carrying. He tells you that he has been
collecting dollars and quarters and they add up to seventeen dollars in value. How many of each coin
does he have?
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MIXTURE PROBLEMS

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Mixture problems occur in many different situations. For example, a store owner may wish to combine two goods
in order to sell a new blend at a given price. A chemist may wish to obtain a solution of a desired strength by
combining other solutions.
Mixture problems are word problems where items or quantities of different values are mixed together. They
involve creating a mixture from two or more things, and then determining some quantity (percentage, price, etc)
of the resulting mixture.
1st mixture + 2nd mixture = total mixture
There are two types of mixture problems:
those in which the items remain separate
(ex. quantities of different costs are mixed together : two different types of candy)
The format for mixture problems involving quantities of different
costs is: [1st amount * 1 st costl + [2n amount * 2 nd cost] = [total
amount * final cost]
those in which two elements blend
(ex. different liquids are mixed together changing the concentration of the mixture: two different types of
chemicals).
The format for mixture problems involving percentages is:
st percentage) + 12 amount * 2nd percentage) = [total amount *final percentage)
[1 amount *
A good way to solve problems like this would be to look at the seoarate mixture types first and begin with a table.
This, however, may take some time and may not be practical during entrance exams, Try to come up with
shortcuts as you get used to solving mixture problems.
MIXTURE PROBLEMS INVOLVING PERCENTS
The key point In mixture problems is In terms of their component parts. For instance, 50cc of 40% alcohol solution
contains more pure alcohol solution than IOOcc of 15% alcohol solution. This is because:
50cc of 40% alcohol solution = 20cc of pure alcohol, and
100cc of 15% alcohol solution = 15cc of pure alcohol.
EXAMPLE:
Forty liters of a 60% salt solution are reduced to a 45% solution. How much solution must be drained off and
replaced with distilled water so that the resulting solution contains only 45% solution.
SOLUTION:
Let x represent the number of liters of salt solution to be drained off and replace with the distilled water.
The content of the mixture is given in the following diagram:
60% salt 60% salt 0% salt 45% salt
40% water 40% water 100% 55% water
40 Liters x Liters water 40 Liters
x Liters

IIS I

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The equation can be based on salt or water solution.
Based on salt solution, the equation is:

Solving for x, we get X = 10 liters

Based on water solution. the equation is:

Solving for x, we get x = 10 liters How much of a 40% solution of alcohol and how much of a 80%
solution should be mixed to give 40 gallons of a 50% solution?

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MIXTURE PROBLEMS INVOL VING QUANTITIES OF DIFFERENT COSTS
The format for mixture problems involving quantities of different costs is:
[1 st amount * 1 st cost] + [2 n amount * 2nd costl = [total amount * final cost]
EXAMPLE: Let's suppose that you work in the grocery store. Suppose macadamia nuts cost $12 per pound and are
not selling well because they are too expensive. You want to mix it with cashews that cost $7 per pound. How
many pounds of cashews should you combine with 25 pound of macadamia nuts to obtain a mixture that can be
sold for $9 per pound?
After you read the problem entirely and get a feel for the whole problem, you first need to set up the
table or use bucket method

Amount Cost in $ Total in $

m.
nuts
cashews

Mixture

Once you have the table or the buckets, you need to fill in the information.
This is how the table will look: This is how the buckets will look:

Amount in pounds Cost in $

25 = 25+C
m. 25 12 $12 $7 $9
nuts
cashews c
Mixture 25+C

Now, let's use the information in the table or in buckets to write the equation.
Remember that amount in first mixture plus amount in second mixture equals amount in total mixture, i.e. 1st
mixture + 2nd mixture = total mixture.
So the equation will be: 12(25) + 7C = 9(25 + C) .
The solution of this equation is C = 37.5pounds.
This means that you should use 25 pounds of macadamia nuts costing $12 per pound and 37.5 pounds of cashews
costing $7 per pound to obtain 62.5 pounds of mixture costing $9 per pound.
Example: How many pounds of candy worth $1.20 a pound must be mixed with 10 pounds of candy worth 90
cents a pound to produce a mixture worth $1.00 a pound?
117 1
How many pounds of candy worth $1.20 a pound must be mixed with 10 pounds of candy worth 90
cents a pound to produce a mixture worth $1.00 a pound?

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MOTION PROBLEMS
A body is moving in a path of constant speed if that body passes over equal distances in any two equal intervals of
time. Such motion is referred to as uniform motion in a path. For example, if the average velocity is 50 miles per
hour (mph), then the distance covered in 2 hours is 100 miles.
The speed or rate of the body in its path is defined as the distance travelled in one unit of time. If r is the
rate and d is the distance travelled in t units of time, then these three quantities are related by the equation:
Distance = rate (time) or d = rt
From this equation, we can derive the associated relationships:
dt
EXAMPLE:
Mrs. Altares boarded a bus bound for Baguio at exactly 4am. After one and one-half hours, her husband followed
driving the family car at 75mph. Assuming that the bus is travelling at an average speed of 50mph, at what time will
Mr. Altares overtake the bus?
SOLUTION:
Let x = the number of hours required for the car to overtake the bus, and
X 1.5 = the number of hours traveled by the bus before overtaking.
t d
CAR 75 751
BUS 50 x + 1.5 50 x + 1.5
For the car to overtake the bus, their distances should be equal, thus, we have the equation:
75x = 50(x + 1.5)
Solving for x, we get x = 3 hours. If the car left at 5:30am, then the car will overtake the bus at 8:30am.
ANOTHER SOLUTION:
Let x = the number of hours covered by the bus in travelling before overtaking takes place. And x
— 1.5 = number of hours covered by the car in travelling
We represent the givens as:
d
CAR 75 x- 1.5 75 x — 1.5
BUS 50 x sox

And we form the following equation:

sox = 75(x — 1.5)
Solving for x, we get x = 4.5 hours. This means that it will take the bus 4.5 hours before overtaking. Since the bus
left at 4am, then the car will overtake the bus at 8:30am.
Two cars leave Monumento at the same time and travel in the opposite direction. If one travels at 62 kph
and the other at 88 kph, how long will it take them to be 750 km apart?
WORK PROBLEMS
Work problems generally involve the portion of a job that a person can do in a given amount of time.
The guiding principle in work problems is the fact that the product of rate of work and the number of units of time
involved is equal to the fractional amount of the work done, i.e.
amount of work done
rate of work =
actual time to finish the work
Or
(rate of work)(actual time to finish the work) = amount of work done
Often, the problems involve two or more workers (say A and B), each having different rates of work to combine
their efforts to complete a job. In such cases, the basic equivalence relationship is
Amount of Amount of Amount of
work done by work done by work done
together

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EXAMPLE:
Jhun can finish an accounting work in 8 hours. Leo can finish the same work in 6 hours. After 2 hours of
working together, Jhun left for lunch and Leo finished the job. How long does it take Leo to finish the
job?
SOLUTION:
Let x represent the number of hours it takes Leo to finish the job.
1

Each worked for two hours. In that time each did two times the fractional part of their work, or 2 (— 8) and 2G)
Jhun left and only Leo worked and finished the job in x hours, so he did x of the job. When the job is completed, all
the fractional parts of the job sum up to one complete job.
Time to finish Work finished in 1 Fractional part Work done in
the work hour done in 2 hours x hours
Jhun 8 hours 1
Leo 6 hours
1
We form the equation:

2
Upon solving x = 2—, X = 2— hours, the number of hours it takes Leo to finish the work alone.
Two cars leave Monumento at the same time and travel in the
opposite direction. If one travels at 62 kph and the other at 88 kph,
how long will it take them to be 750 km apart?

121 1
A swimming pool can be filled in 6 hours and requires 9 hours to drain. If the drain was accidentally left
open for 6 hours while the pool was being filled, how long did the pool require to fill?
AGE PROBLEMS
This type of problem usually relates the ages of two persons. Often encountered is the word "ago" which means
that the previous years. On the other hand, the phrases "x years hence", 'tx years from now", or "in x years" all
refer to x years to come or x years in the future.
EXAMPLE:
Lita is 41 years old and her daughter is 9, In how many years will the mother be three times as old as her daughter?
SOLUTION:
Let x = the required number of years
9 + x = daughter's age after x years
41 — x = Mother's age in x years
The equation is based on:
Mother's age in x years = 3(daughter's age in x years)
41 X = X) x =
7years

The sum of the ages of the two brothers is 20. In five years, one is twice the other. Find the present ages
of the two brothers.

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PROBLEM SOLVING EXERCISES A

1. Number
Find three consecutive numbers whose sum is 60.
2. Age
Paolo is 5 years older than Ryan. Two years ago, Paolo is four times one-third of Ryan's age. What is Ryan's
age now?
3. Money / Coin
I have P35 consisting of one-peso and five peso coins. There were 5 more one-peso than five-peso coins.
How many are there of each kind?
4. Mixture
A solution consists of 20% acid and the remainder is water. A chemist wants to strengthen this solution. How many
liters of acid must be added to 6 liters of the solution to make a 40% mixture?
5. Work
A boy can do a job in 2 hours, while his brother can do it in I % hours. How long will it take to do the job if
they work together?
6. Motion
Two men leave at 6:00 am. from towns 40 kilometers apart and walk toward each other. The first walks at an
average rate of 3 % kilometers per hour and the second at an average rate of 4 h kilometers per hour. At
what time will they meet?
Page
Current
A boat can travel 8 kilometers per hour in still water. If it can travel 20 kilometers downstream in
the same time that it can travel 12 kilometers upstream, what is the rate of the stream?
8. Compound Interest
a) Find the compound amount at the end of two years on an original principal of P5000 at 6%
compounded annually.
b) If the P5000 were invested for two years at 6% compounded semi-annually, what would be the amount?
9. Investment
An man invests P35,OOO part at 3% and part at 4%. How much does he invest at each rate if he receives
interest totaling PI, 150 in one year?
10. Clock
If it is now 12:00 noon, at what time before 1:00 pm. would the hands of the clock be perpendicular with
each other for the first time?

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 Find the area. writing the answer as a polynomial in descending powers of x.
3X+9
Angela pays $356 in advance on her account at the athletic club. Each time she uses the club,
$9 is deducted from the account. Find a linear function that models the value remaining to the
club. Find the remaining value in the account after 15 visits.
A rectangle has length x+9 and width x-1. Find the equation that describes the area, A, of the
rectangle in terms of
x.
The cost of 40 shares of Fly-by- Night Airlines is $57.50. How many shares can you buy with
$575.00?
A certain sum of money is invested in a business. In each year, this investment earns 1 1/ times as much
as in the preceding year. If the investment earned a total of $29,250.00 in four year, how much did it
earn in the fourth year?
Sandy draws a sequence of circle. She starts with a row of 4 circles. The second row has 2 more circles
that the first row, the third row has 2 more circles than the second, and so on. How many circles does
she have in the tenth row?
The price per person of renting a bus varies inversely with the number of people renting the bus. It costs $34 per
person if 64 people rent the bus. How much will it cost per person if 81 people rent the bus?
Geothermal energy is heat from inside earth. Underground temperatures generally increase 9 0Cfor every 300
meter of depth. How deep would a well have to be for the temperature to reach 162 0C 1620C? (Assume surface
temperature is O O C OOC)
A piece of machinery valued at $60,000 depreciates $5,000 the first year, $4,800 the second year, $4,600 the third
year, and so on. Find the value of this piece of machinery at the end of six years.
The sum of the reciprocal if a number and the reciprocal of 3 less that the number is 8 times the
reciprocal of the original number. Find the original number.

Chase earned $16 Monday and doubled his pay each day thereafter. How much did he earn on Friday?
|
The amount of oil used by a ship travelling at a uniform speed varies jointly with the distance and the square of
the speed. If the ship uses 500 barrels of oil in travelling 400 miles at 44 mph, determine how many barrels of oil
are used when the ship travels 440 miles at 22 mph
x+28 *+2B
The production rate of a small factory is modelled by— while the production rate of another factory
sx(x+2)'
9X+ 14 9+14

is modelled by . What would be a model for the combined production rate of the two factories?
Ziggy has a stock of 4 index cards. With a pair of scissors he cuts the stack in half and then places all the resulting
card pieces in a new stack. If he does this a total of four times, how many card pieces will he have?
11
A rubber ball dropped on a hard surface take a sequence of bounces, each one — as high as preceding one, if this
22 ball is dropped form a height of 12 feet, how far will it have travelled when it hits the surface the fifth time?
Kirk's average driving speed is 10 miles per hour faster than Mia's. In the same length of time it takes
Kirk to drive 354 miles, Mia only drives 294 miles. What is Kirk's average speed?
The length of a rectangle is 12 centimetres less than three times its width. If the area is 36 square centimetre, find
the length and width

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The product of two consecutive positive integers is 119 more than the next integer. What is the largest of the
three integers?
A juice drink manufacturer uses fresh squeezed juice and cherry juice concentrate to make its breakfast
beverage. How many gallons of 40% cherry juice must be mixed with 60 gallons of 13% cherry juice to obtain a
drink that is 30% cherry juice?
One leg if right triangle is 17 inches longer than the other leg. The hypotenuse is 25 inches long. How long is the
shorter leg?
How much pure water must be mixed with S pints of 50% developer to produce a mixture that is 11% developer?
The sum if the digits of a two digit number is eight. If the digits are reversed, the new number is eighteen less than
the original number. Find the original number.
A motorboat can go 8 miles downstream on a river in 20 minutes. It takes 30 minutes for this boat to go back
upstream the same 8 miles. Find the speed of the boat.
A jar filled with only nickels and quarters contains a total of 48 coins. The value of all the coins in the jar is $7.60.
How many quarters are in the jar?
Tom is 5 times as old as Todd. In 2 years, Tom's age will be 12 years more than 3 times as old as Todd. How old is
Todd?
131 1

age
Mathematics 2014 09155057703 3998584