MULTIPLYING AND DIVIDING FRACTIONS The Number System 1 © 2013 Meredith S. Moody

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MULTIPLYING AND DIVIDING FRACTIONS

The Number System

© 2013 Meredith S. Moody

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Objective: You will be able to…

Convert fractions to their reciprocals and back

Divide a fraction by another fractionSolve problems by dividing fractions by

fractions


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The Fraction

When we divide a whole into equal parts, we represent those parts as fractions


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Part of a Part

Why would we need to divide a fraction by a fraction?

We may not have a whole number to divide into pieces

How much is half of a half? What about a third of a half? What about a half of a fourth?


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Example 1

Let’s say you have a cake for a partyThe day of the party, you serve half of the

cake to your guests, but have half left overThe next day, you have some friends over and

you want to serve the other half of the cake, but you want everyone to have the same number of pieces

Here, you would have to divide a fraction ( ½ of the cake) by a whole number (the number of friends you have over)


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Example 1, visual

Drawing pictures is helpful when dividing a fraction by a fraction

You start with a whole, and then divide it in half

1 whole cake ½ of the cake


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Example 1, visual, continued

When half the cake is gone, you only have the leftover half to work with

If you have 1 friend over, you have to split the half into 2 pieces (1 for you, 1 for the friend)

½ of a ½


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Example 1, visual, continued

You can look at how the smallest piece fits into the whole:

½ of a ½ is equal to ¼ of the whole

½ of a ½


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Example 1, Algebraic translation

What operation represents ‘½’ something?½ of a 1 whole is the same thing as dividing 1

whole by 2This is because the fraction bar represents

the operation of division ( ½ = 1 ÷ 2)So if we want to ‘half’ something, we divide

by 2So, ‘half’ing a half is = ½ ÷ 2½ ÷ 2 = ¼


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Reciprocals

Reciprocal fraction: The result of interchanging the numerator and denominator in a fraction

A whole number can be represented as a fraction over ‘1’ (because a number divided by ‘1’ is itself)

Therefore, the whole number 5 = 5/1 and the

reciprocal = 1/5


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Example 2, visual

Working with halves is easier than other fractions

What if I wanted to know what a third of a half is?

1 whole cake½ of the

cake

1/3 of the ½

1/6 of the whole


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Here, I took ½ and divided by 3 to find 1/3 of the ½

½ ÷ 3 = 1/6½ ÷ 3/1 = 1/6Do you see a pattern yet?


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Example 3

What if I wanted to find something more complicated?

What if the first day of my party, I served 1/3 of the cake, so I had 2/3 left over

The next day, I had 8 friends over, and I wanted each person to have a slice, but three said ‘no thanks’ – so I only served 5/8 of the 2/3

What is the leftover 3/8 of the 2/3?So I want to know: what is 3/8 of 2/3?


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Example 3, visual

Here is the visual representation of 3/8 of 2/3

1 whole cake the cake

2/3 of

is leftwhole

of

cakethe6/243/8 of

2/3 ofthe cake


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I divided 2/3 into 8 pieces 2/3 ÷ 8But I wasn’t interested in just 1 of those

pieces, I wanted to figure out what 3 of them were

I didn’t want 2/3 ÷ 8/1I wanted 2/3 ÷ 8/3So 2/3 ÷ 8/3 = 6/24 Do you see a pattern yet?


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Seeing the Pattern

Let’s review:½ ÷ 2 = ½ ÷ 2/1 = ¼ ½ ÷ 3 = ½ ÷ 3/1 = 1/62/3 ÷ 8/3 = 6/24What is the pattern?That’s right! The answer is the dividend (1st

fraction) multiplied by the reciprocal of the divisor (2nd fraction)!


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Using Reciprocals

We can use reciprocals and our understanding of multiplication to divide a fraction by a fraction

½ ÷ 2 = ½ ÷ 2/1 = ¼ When I divide a ½ by two, I’m finding ½ of

the ½ What operation does ‘of’ represent?That’s right! Multiplication!½ ÷ 2 = ½ ÷ 2/1 = ½ x ½ = ¼


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Check the Pattern

Does this pattern always work?Let’s check our second cake scenario, where

we wanted to find 1/3 of ½ We took ½ the cake and cut it into 3rds and

looked at one of those pieces: ½ ÷ 3 = ½ ÷ 3/1 = 1/6

1/3 of ½ = 1/3 x ½ = 1/6Does the pattern fit?Yes!


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Check the Pattern

Let’s check our last cake scenario, where we wanted to find 3/8 of 2/3

We took 2/3 of the cake and cut it into 8ths and looked at 3 of those pieces

2/3 ÷ 8/3 = 6/243/8 of 2/3 = 3/8 x 2/3 = 6/24Does the pattern fit?Yes!


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Algebraic Rule

Based on the pattern we found, we can write a rule for dividing a fraction by another fraction:

a/b ÷ c/d = a/b x d/c = ad/bc


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You try

Evaluate the following expressions. Remember to write your answer in lowest terms!

½ ÷ ⅛ 8/2 = 4

⅞ ÷ ¼ 28/8 = 7/4


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Mixed Numbers

What if I have mixed numbers as part of my problem?

Convert the mixed number to an improper fraction, then solve


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You try

6 and ¼ ÷ 3 and 5/9 25/4 ÷ 32/9 25/4 x 9/32 225/256

3 and 2/7 ÷ 2 and 5/6 21/2 ÷ 17/6 21/2 x 6/17 126/34 3 and 24/34 3 and 12/17


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MULTIPLYING AND DIVIDING FRACTIONS The Number System 1 © 2013 Meredith S. Moody