Fractions And Decimals

## Lesson 4 mathematical Operations on Fractions

Welcome to Lesson 4 Mathematical Operations on Fractions. It is good that you have completed last three lessons. You are now further building the concepts of Fractions and getting ready to perform Mathematical operations on them.

Objectives: In this lesson, you will know what Least Common Multiple (LCM) of numbers is.

You will know

·         How to add and subtract the Proper Fractions with same denominators?

·         How to add and subtract the Proper Fractions with different denominators?

·         How to perform multiplication and division on Proper Fractions?

·         How to add and subtract the Mixed Fractions with same denominators?

·         How to add and subtract the Mixed Fractions with different denominators?

·         How to perform multiplication and division on Mixed Fractions?

How to add and subtract the Proper Fractions with same denominators?

Fractions consist of two numbers. The top number is called the numerator. The bottom number is called the denominator.

numerator
denominator

Proper Fraction has smaller numerator than the denominator.

·         To add two proper fractions with the same denominator, add the numerators and place that sum over the common denominator.

·         To subtract two proper fractions with the same denominator, subtract the second numerator from the first one and place that subtraction as the numerator over the common denominator.

How to add and subtract the proper fractions with different denominators?

·         Find the Least Common Denominator (LCD) of the fractions

·         Rename the fractions to have the LCD

·         Add/Subtract the numerators of the fractions

·         Reduce/Simplify the Fraction

The Least Common Denominator (LCD) is the Least Common Multiple (LCM) of two or more denominators.

How to find the Least Common Denominator (LCD or LCM):

·         Find the Greatest Common Factor of the denominators.

·         Multiply the denominators together.

·         Divide the product of the denominators by the Greatest Common Factor.

Example: Find the LCD of 2/9 and 3/12

·         Determine the Greatest Common Factor of 9 and 12 which is 3

·         Either multiply the denominators and divide by the GCF (9*12=108, 108/3=36, LCD = 36)

·         OR - Divide one of the denominators by the GCF and multiply the quotient times the other denominator (9/3=3, 3*12=36 LCD=36)

How to rename fractions and use the Least Common Denominator:

·         Divide the LCD by one denominator.

·         Multiply the numerator times this quotient.

·         Repeat the process for the other fraction(s)

·         LCD is 36

·         First fraction (2/9): 36/9 = 4, 4*2 = 8, first fraction is renamed as 8/36

·         Second fraction (3/12): 36/12 = 3, 3*3 = 9, second fraction is renamed as 9/36

·         It is possible to add or subtract fractions that have the same denominator

·         8/36 + 9/36 = 17/36

·         How to perform multiplication and division on proper Fractions?

Multiplication of Fractions:

·         Multiply the numerators of the fractions

·         Multiply the denominators of the fractions

·         Place the product of the numerators over the product of the denominators

·         Simplify the Fraction

Example: Multiply 2/9 and 3/12

·         Multiply the numerators (2*3=6)

·         Multiply the denominators (9*12=108)

·         Place the product of the numerators over the product of the denominators (6/108)

·         Simplify the Fraction (6/108 = 1/18)

·         The Easy Way.  It is often easiest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.

·         For example:

2/9 * 3/12

= (2*3)/(9*12)

= (1*3)/(9*6)

= (1*1)/(3*6)

= 1/18

Multiplying the Proper Fraction by an Integer.

·         Multiplying a fraction by an integer follows the same rules as multiplying two fractions.

·         An integer can be considered to be a fraction with a denominator of 1.

·         Therefore when a fraction is multiplied by an integer the numerator of the fraction is multiplied by the integer.

·         The denominator is multiplied by 1 which does not change the denominator.

For Example:

2/15 * 3

= 2/15 * 3/1

= 2/5

Division of Proper Fractions

·         Take the reciprocal of the Fraction which is dividing the first fraction, (i.e. Turn over the Fraction -the denominator becomes numerator and numerator becomes denominator )

·         Multiply the two fractions.

·         Multiply the numerators of the fractions

·         Multiply the denominators of the fractions

·         Place the product of the numerators over the product of the denominators

·         Simplify the Fraction

Example: Divide 2/9 and 3/12

·         Reciprocal of the second fraction is 12/3

·         Multiply two fractions (2/9 ÷ 3/12 = 2/9 * 12/3)

·         Multiply the numerators (2*12=24)

·         Multiply the denominators (9*3=27)

·         Place the product of the numerators over the product of the denominators (24/27)

·         Simplify the Fraction (24/27 = 8/9)

·         The Easy Way.  After inverting, it is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.

·         For example:

2/9 ÷ 3/12

= 2/9*12/3

= (2*12)/(9*3)

= (2*4)/(3*3)

= 8/9

Division of Fractions by Integer

·         Treat the integer as a fraction (i.e. place it over the denominator 1)

·         Invert (i.e. turn over) the denominator fraction and multiply the fractions

·         Multiply the numerators of the fractions

·         Multiply the denominators of the fractions

·         Place the product of the numerators over the product of the denominators

·         Simplify the Fraction

Example: Divide 2/9 by 2

·         The integer divisor (2) can be considered to be a fraction (2/1)

·         Invert the denominator fraction and multiply (2/9 ÷ 2/1 = 2/9 * 1/2)

·         Multiply the numerators (2*1=2)

·         Multiply the denominators (9*2=18)

·         Place the product of the numerators over the product of the denominators (2/18)

·         Simplify the Fraction if possible (2/18 = 1/9)

·         The Easy Way.  After inverting, it is often simplest to "cancel" before doing the multiplication. Canceling is dividing one factor of the numerator and one factor of the denominator by the same number.

·         For example:

2/9 ÷ 2

= 2/9 ÷ 2/1

= 2/9*1/2

= (2*1)/(9*2)

= (1*1)/(9*1)

= 1/9

Mathematical Operations on Improper Fractions/Mixed Fractions.

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