This document provides an overview of topics covered in a mathematics guide for civil service examinations. It includes: 1) classification of numbers like rational, irrational, integers; 2) operations on numbers such as addition, subtraction, multiplication, division of signed numbers; 3) concepts like prime factorization, LCM, GCF, ratios, proportions, percentages; 4) geometry, algebra, sets, probability, series; and 5) measurement conversions between metric units. Worked examples are provided to illustrate key rules and procedures.

1. MATHEMATICS
Ultimate Learning Guide for Civil Service
Examination

2. Components:
I. Classification of Numbers
II. Operation on Numbers
III. Plane Geometry
IV. Algebraic Expression
V. Sets Operation
VI. Probability and Statistics Problem
VII.Series and Sequence Problems
VIII.Word Problems

3. Pre-Test:
Identify the given numbers whether it is rational or irrational number.
1. Π
2. 0
3. -2.1
4. 7
5. 1. 287645

4. I. CLASSIFICATION OF NUMBERS
Rational Irrational
(Non-Terminating and Non-Repeating decimals)
Fractions / Decimals Integers
Negative Integers Whole Numbers
Zero Natural Numbers

6. II. OPERATION ON NUMBERS
A. Addition and Subtraction of Sign Numbers
Illustration 1:
1. 9 - 6 =
2. -9 + 6 =
3. 9 + 6 =
4. -9 - 6 =
Illustration 2:
1. 7 - 5 + 3 - 1 =
RULES:
Like sign: (Add then copy the sign)
Unlike sign: (Subtract then copy the sign)
3
4
-3
15
-15

7. B. Multiplication and Division of Sign Numbers
Illustration 1:
1. 6 (9) =
2. -6 (-9) =
3. 6 (-9) =
4. -6 (9) =
Illustration 2:
RULES:
Like sign (positive)
Unlike sign (negative)
54 5
54
-54
-54
5
-5
-5
2. -10 =
-2
1. 10 =
2
3. -10 =
2
4. 10 =
-2

8. C. Prime Factorization
● It is the process of finding which prime numbers you need to multiply
together to get the original number.
● In the factor tree method, we
use a pictorial method of finding
factors, where the numbers to be
factorized is placed at the top and
all its factors branch out one by
one till we get all the prime factors
, just like a tree.

9. ● In the continuous division method , we perform repeated divisions using
prime factors as divisors until the last dividend becomes 1.

10. LCM AND GCF
● LCM is the smallest multiple that 2 or more numbers have in common.
For example, L.C.M of 16 and 20 will be 2 x 2 x 2 x 2 x 5 = 80, where 80 is the
smallest common multiple for numbers 16 and 20.
Now, if we consider the multiples of 16 and 20, we get;
16 → 16, 32, 48, 64, 80,…
20 → 20, 40, 60, 80,…,
We can see that the first common multiple for both numbers is 80. This proves
the method of LCM is correct.

11. GCF
● GCF is the largest multiple that can exactly divide 2 or more numbers.
● GCF (16, 20) = 2 x 2 = 4
● 16 = 1, 2, 4, 8
● 20 = 1, 2, 4, 5, 10
● Remember: For GCF, after you list down the prime factors, mark similar
prime factors as one pair. Then multiply the paired prime factors ONLY.

12. Divisibility Rules
Divisor Divisibility Condition
2 The last digit is even (0,2,4,6,or 8).
3 Sum of the digits is clearly divisible by 3.
4 The last two digits are divisible by 4.
5 The last digit is 0 or 5.
6 It is divisible by 2 and by 3
7 Subtract 2 times the last digit from the rest. The result should be
divisible by 7.
8 The last three digits are divisible by 8.
9 Sum of the digits is divisible by 9.
10 The last digit is 0.

13. Ratio and Proportion
● Ratio is an expression of the relative size of two quantities.
It is usually expressed as the quotient of one number divided by the other.
The ratio of 1 to 2 is written as 1:2 or 1/2
● Proportion is a statement of equality between two ratios.
The ratio of 1:2 to 3:6 forms the proportion
1:2 = 3:6 or 1/2 = 3/6

14. Example:
1. (CSE, March 2018) Delegates from Luzon, Visayas and Mindanao will be
chosen in the ratio of 4:3:2, respectively. If Luzon has 120 delegates, what is the
total number of delegates?
1. 60 2. 90 3. 120 4. 240 5. 270
Solution:
Here, Luzon: Visayas: Mindanao = 4:3:2 .
To get the multiplier, we need to divide 120 by 4. That is 120/4 = 30.
This means that 4:3:2 = 4(30):3(30):2(30)
Therefore 4:3:2 = 120:90:60
The total number of delegates = 120+90+60= 270

15. Percentage
● In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100.
● In problems, it is the number that comes before the word is.
● If we have to calculate percent of a number, divide the number by the whole and multiply by 100.
Hence, the percentage means, a part per hundred. The word per cent means per 100. It is
represented by the symbol “%”.
Examples of percentages are:
● 10% is equal to 1/10 fraction
● 20% is equivalent to 1/5 fraction
● 25% is equivalent to ¼ fraction
● 50% is equivalent to ½ fraction
● 75% is equivalent to ¾ fraction
● 90% is equivalent to 9/10 fraction
● Percentages can also be represented in decimal or fraction form, such as 0.6%, 0.25%, etc.

16. BASE
● It is the number or quantity which represents the original
number.
● It also represents the total.
● It is obtained by dividing the percentage by the rate.
● In problems, it is the number that comes after the word of.
RATE
● It is the number that represent the percent.
● It is obtained by dividing the percentage by the base.
● In problems, it is the number that is attached to the word percent
or to a % sign.

17. Example:
Problem 1. What is 12.5% of 150?
1. 1,875 2. 187.5 3. 18.75
4. 37.5 5. 12.5
Solution: Note that 12.5 % converted to decimal form is 0.125. Since the given
number e is 150 and were looking for t:
t = ae
t = (0.125)(150) = 18.75
Let t = percentage
a = rate( must be converted to decimal form)
e = base
Note that in percentage
problems the word “is”
means equals and the
word “of” means multiply.
Therefore: t = ae
a = t/e
e = t/a

18. Order of Operation
1. Parenthesis
2. Exponents
3. Multiplication
4. Division
5. Addition
6. Subtraction
Example:
1- 2{5-2(2-4)^2}
= 1-2{5-2(-2)^2}
= 1-2{5-2(4)}
= 1-2 {5-8}
= 1-2{-3}
= 1+6
= 7

19. FRACTIONS
Similar Fraction
(the same deniminator)
e.g. 1/8, 3/8, 5/8
Improper Fraction
(fraction whose numerator is greater than the denominator)
e.g. 7/4, 5/2, 9/5
Proper Fraction
(fraction whose denominator is greater that the numerator)
e.g. 2/5, 3/7, 1/9
Equivalent Fraction
(fraction with the same value)
e.g. 2/8 = 1/4

20. Simplifying Fraction
●
- In reducing to lowest term, divide the numerator and the
denominator by the GCF.
ac = a
bc b
Example:
12 = 2 · 6 = 2
18 3· 6 3

21. ● If the fractions are similar, copy the denominator then perform the indicated
operations on the numerator.
a/b + c/b = a+c Example: 2/5 +1/5 = 3/5
b
● If not similar, a/b + c/d = ad + bc
bd
Example: 2/9 - 1/4 = 2(4) - 1(9) = 8-9 = -1/36
9(4) 36

22. ● Multiplication
a/b · c/d = ac/bd
● Division
a/b ÷ c/d = a/b · d/c = ad/bc

23. MEASUREMENT CONVERSION
-It is done to determine equivalent SI units using different prefixes.
For example, to convert 8 kilometers (8km) into centimeters,
8km 10^3 m 1cm = 8 x 10^5 cm or 800,000 cm
1km 10^-2m

24. SI Prefixes for Multiples
SI Prefixes for Fractions
Name deci centi milli micro nano pico femto atto zepto yocto
Symbol d C m µ n p f a z y
Factor 10^-1 10^-2 10^-3 10^-6 10^-9 10^-12 10^-15 10^-18 10^-21 10^-24
Name deca hecto kilo mega giga tera peta exa zetta yotta
Symbol da H k M G T P E Z Y
Factor 10 10^2 10^3 10^6 10^9 10^12 10^15 10^18 10^21 10^24

25. Metric Unit Conversion
Example:
7m = 7000 mm
3km = ____m
5km = ____dam
km hm dam m dm cm mm
7 0 0 0