Basic Mathematics

Prepared by

RONALDO Z. ONGOTAN

Competencies:

1. Properties of Numbers2. Numbers Theory3. Integers4. Fraction and Decimals5. Operation with whole numbers,

decimals, and fractions6. Ratio and Proportion7. Percent

PROPERTIES OF NUMBERSA. SET OF NUMBERS

Real Numbers (R)-9, 6, , 0, 1, , 21, 5 2

314

Rational Numbers (Q)Irrational Numbers (H)-9, , 0, 1, , 5 2

314 6, 21, 5

Integers (Z) Non-Integers…-3, -2, -1, 0, 1, 2, 3,

23

14,

Negative Integers Whole Numbers (W)…-3, -2, -1, 0, 1, 2, 3,

Zero Natural Numbers (N) 1, 2, 3….0

Numbers that can be expressed as the quotient or ratio of two integers a and b, represented as a, where b≠0 bHave a specific place on the number line.Can be written as terminating ( e.g. 1.75, 2.5) or repeating decimals (e.g.0.1111…,2.090909)

THE REAL NUMBER SYSTEM

 Subsets

 Definition

 Natural

Numbers (N)

  Consist of the numbers 1, 2,3…

 Whole Numbers

(W)

  Consist of the Natural numbers

zero 

Integers  Consist of the natural numbers,

their negatives, and zero

B. THE ORDER OF OPERATIONS

When performing multiple operations, remember PEMDAS:

P parentheses (grouping symbols)E exponentsMD multiply and divide from left to rightAS add and subtract from left to right

 Property

 Definition

 Examples

 Closure Property

When we add or multiply any element in a set of numbers, the sum or product is a unique real number which belongs to that same set.

14+25+44+52+135 23×14×10= 3 220

 Commutative

Property

States that changing the position of the addends or the factors does not affect the sum or the product.

33×10=10+33 18×9= 9×18

 Associative Property The sum of any number and

zero Addition is the same number.

•0 is the identity

(5+6) + 7=5+ (6+7) 2× (4×6)= (2×4)×6

 Identity Property The product of any number

and one isMultiplication the same number.

1 is the identity.

 512×1 = 512

Distributive Property of Multiplication over Addition/Subtraction

states that multiplication distributes over addition and subtraction

7(9+12)=7(9)+7(12)

C. NUMBER PROPERTIES

II. NUMBER THEORY

A. DIVISIBILITY RULES

B. FACTORS AND MULTIPLES

The Factors of integers n are the positive integersThat divide n evenly without remainder.

Ex: factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The multiples n are the integers that n divides withoutAny remainders.

Ex: multiples of 7: 7,14,21,28,35...

C. PRIME AND COMPOSITE NUMBERS

Prime Number- counting numbers that have exactly two distinct, positive divisors

e.g. 2,3, 5, 7, 11, 13, 15, 17, 19…

Composite Number- counting numbers greater than 1 thathave positive factors other than 1 anditself

e.g. 4, 6, 8, 9, 10, 12, 14, 16…

Prime Factorization

Expressing a number as a productOf factors, each of which is a primeNumber.

Methods in Finding the Prime Factors of a Given Number

Factor Tree Method Continuous Division Method

D. GCF and LCMGREATEST COMMON FACTOR Refers to the largest

common factor of two or more numbers

D. GCF and LCMLEAST COMMON MULTIPLE Refers to the smallest

number thatTwo or more numbers

will divide Without remainder

III- INTEGERSINTEGERS Refer to the set of whole number and opposites

ABSOLUTE VALUE The number of units a number is away

from 0 in a number line

INTEGER OPERATIONS

TANDAAN!!!

Kung MINUS ang sign that is UTANGKung PLUS ang sign that is BAYAD

IV. FRACTIONS Fractions

A number whose value can be expressed as the quotient or ratio of any two numbers a and b, represented as a, where b≠0. It is a part of a whole or a set. Reducing Fractions to Lowest TermsDivide the numerator and the denominator by its GCF,Example:

18÷6 = 324 6 4

CONVERSIONS

A. Mixed Number To Improper Fraction

B. Improper Fraction ToMixed Number

FRACTIONS OPERATIONS 

Addition and SubtractionOf Similar Fractions:Just add/subtract the numerators and copy the denominator.Examples:

2 + 4 = 6 7 7 7Of Dissimilar Fractions:Convert the fractions first to similar fractions. Then add/subtract the numerators and keep the denominators. Reduce to lowest terms if necessary.

B. Multiplication of FractionsC. Division of Fractions

V. DECIMALSA.CONVERSIONS

1. Decimal to Fraction / Mixed Number2. Fraction to Decimal3. Mixed Number to Decimal

B. DECIMAL OPERATIONS

a) Addition and Subtraction b) Multiplication c) Division

 Ratio

 Rate

 Proportion

A comparison of two or more amounts or quantities, such as a and b, can be expressed in the following equivalent ways: a:b, a/b, a b 

Refers to a ratio whose two amounts represent different quantities Examples: 35mi/h’ 5m/s

   

An equation or statement that expresses the equality of two ratios.May be expressed as:1) a =c, or

b d2) a: b = c:dIn each form, b and c are called means, and a and d are called extremes.

VI. RATIO AND PROPORTION

TYPES OF PROPORTIONS

A. Direct ProportionAs one quantity increases, the other also increases.

B. Inverse ProportionAs one quantity increases, the other quantity decreases, and vice versa.

C. Partitive Proportion One quantity is being partitioned into different

proportions.

Example:A piece of wood 150cm long is cut in the ration

2:3:5. Find the measure of each part.

APPLICATIONS INVOLVING RATIOS AND PROPORTIONS

A.Scales

When working with scale models, the scale is often given as the ratio:

Example:If the scale model of a boat measures 6 inches and the model has a scale of 1:20, what is the actual measurement of the boat?Solution: model length = 1 = 6

Actual length 20 x 

1 = 6 20 x

x=120Answer: The actual measurement of the boat is 120 in., or 10 ft.

A10 cm

B C6 cm

x

15 cm

D

E F

Solution:

AB = BC 10 = 6DE EF 15 x  10x = 90

x = 9 

B. Similarity•When figures have corresponding sides that are in proportion with one another and corresponding angles with the same measure, the figures are similar.•Proportion can be used to determine that figures are similar, and calculate the missing part/s of known similar figures.Example:

Find the missing side of the larger triangle.

 Between Decimals and Percent

 Between Fractions and Percents

A. Percent to Decimal Remove the percent symbol

(%) and move the decimal point two places to the left.

A. Decimal Percent1. Multiply the decimal by 100;

or2. Move the decimal point two

places to the right and write a percent symbol.

A. Percent to Fraction Use the proportion x = a, and

cross multiply 100 bto solved for the variable x. 

A. Fraction to Percent Remove the percent symbol (%)

and multiply the number by 1 100

VII. PERCENTSPercent•Literally meaning ‘per hundred’, it refers to a special ratio that compares a numerical quantity to 100.•CONVERSATIONS

APPLICATION INVOLVING PERCENT

A. Percent Increase or Decrease

To increase a number by a certain percent, (1) add 100% to the given percent, (2) convertthe sum to a decimal, and (3) multiply the number by that decimal. Example: increase 40 by 45%Solution: 45% + 100% = 145% =1.45 40 x 1.45 = 58

B. Simple Interest Rate

I = Prt I interest charged or paid outP principle amount that is saved or borrowedr percentage rate written as a decimal

t time in years

Example:

If Cedric borrows P15,000 at an interest rate of 17% for 18 months, how much will he have paid in simple interest at the end of the 18 months? Solution: I=Prt

I=(P15,000) (17%) (1.5)I= P3,825

P P15,000

r 17%t 18 months = 1.5 years

Thank you