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Basic Mathematics
Prepared by
RONALDO Z. ONGOTAN
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Competencies:
1. Properties of Numbers2. Numbers Theory3. Integers4. Fraction and Decimals5. Operation with whole numbers,
decimals, and fractions6. Ratio and Proportion7. Percent
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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
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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
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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
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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
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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
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II. NUMBER THEORY
A. DIVISIBILITY RULES
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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...
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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…
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Prime Factorization
Expressing a number as a productOf factors, each of which is a primeNumber.
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Methods in Finding the Prime Factors of a Given Number
Factor Tree Method Continuous Division Method
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D. GCF and LCMGREATEST COMMON FACTOR Refers to the largest
common factor of two or more numbers
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D. GCF and LCMLEAST COMMON MULTIPLE Refers to the smallest
number thatTwo or more numbers
will divide Without remainder
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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
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TANDAAN!!!
Kung MINUS ang sign that is UTANGKung PLUS ang sign that is BAYAD
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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
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CONVERSIONS
A. Mixed Number To Improper Fraction
B. Improper Fraction ToMixed Number
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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.
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B. Multiplication of FractionsC. Division of Fractions
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V. DECIMALSA.CONVERSIONS
1. Decimal to Fraction / Mixed Number2. Fraction to Decimal3. Mixed Number to Decimal
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B. DECIMAL OPERATIONS
a) Addition and Subtraction b) Multiplication c) Division
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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
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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.
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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.
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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.
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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
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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
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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
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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
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Thank you