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Dividing Fractions by Fractions
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 We'll learn how to divide fractions by fractions.
 We will also learn how to divide mixed numbers by mixed numbers.
 And we will see how this knowledge can help us throw a party, go for a run, and even make our own clothes!

Discussion Questions

Before VideoWhat happens if you divide 1 pie by 4? What happens if you divide it by [ggfrac]1/4[/ggfrac]?ANSWER

If I divide 1 by 4, I am sharing 1 pie among 4 people, and each person gets [ggfrac]1/4[/ggfrac] of the pie. If I divide a pie by [ggfrac]1/4[/ggfrac], then I want to know how many [ggfrac]1/4[/ggfrac]sized pieces there are in the pie. There are 4.

To multiply fractions, I multiply the numerators together and multiply the denominators together. Therefore, [ggfrac]1/4[/ggfrac] x [ggfrac]5/6[/ggfrac] = [ggfrac]5/24[/ggfrac].

I want to know how many groups of [ggfrac]2/3[/ggfrac] there are in 4 wholes. If I draw a model of the problem, I split each of 4 whole circles into 3 equal parts, and put 2 of those parts into each group. I count 6 groups of [ggfrac]2/3[/ggfrac]. Therefore 4 ÷ [ggfrac]2/3[/ggfrac] = 6. If I multiply 4 x [ggfrac]3/2[/ggfrac], I get [ggfrac]12/2[/ggfrac] = 6, so 4 ÷ [ggfrac]2/3[/ggfrac] is the same as 4 x [ggfrac]3/2[/ggfrac]. I keep the first number the same, change the operation to multiplication, and flip the fraction.

My answer is more than 3. If I divide by a number that is less than 1, that means I am making many little pieces and I want to know how many groups there are. If I divide 3 wholes into pieces of size [ggfrac]1/5[/ggfrac], I count 15 groups of [ggfrac]1/5[/ggfrac].

I can split 2 into fourths as indicated in the denominator, 4. So each of the 2 wholes is split into 4. I have 8 fourths, plus the 3 fourths from [ggfrac]3/4[/ggfrac]. In total, there are 11 fourths, so 2[ggfrac]3/4[/ggfrac] = [ggfrac]11/4[/ggfrac].


After VideoHow can you turn [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/2[/ggfrac] into a multiplication problem?ANSWER

I change the operation to multiplication and flip the second fraction. [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/2[/ggfrac] = [ggfrac]3/4[/ggfrac] x [ggfrac]2/1[/ggfrac].

No, I must keep the first fraction the same and flip the second fraction. The first fraction tells how much I have, and the second fraction tells the size of the groups I am trying to make.

No! To solve [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/2[/ggfrac], I want to know how many onehalfsized pieces there are in threefourths. [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/2[/ggfrac] = [ggfrac]3/4[/ggfrac] x [ggfrac]2/1[/ggfrac] = [ggfrac]3/2[/ggfrac] = 1[ggfrac]1/2[/ggfrac]. To solve [ggfrac]1/2[/ggfrac] ÷ [ggfrac]3/4[/ggfrac], I want to know how many threefourthsized pieces there are in one half. [ggfrac]1/2[/ggfrac] ÷ [ggfrac]3/4[/ggfrac] = [ggfrac]1/2[/ggfrac] x [ggfrac]4/3[/ggfrac] = [ggfrac]2/3[/ggfrac]. I get two very different answers!

I need to turn 1[ggfrac]3/5[/ggfrac] into a mixed number. 1[ggfrac]3/5[/ggfrac] = [ggfrac]8/5[/ggfrac].

It should be less than [ggfrac]1/2[/ggfrac], because the number that I am dividing by, 1[ggfrac]3/5[/ggfrac] or [ggfrac]8/5[/ggfrac], is greater than 1.



Vocabulary

Fraction
DEFINE
A number that expresses equal parts of a whole.

Numerator
DEFINE
The number on the top of the fraction, which tells us how many parts of the whole we have.

Denominator
DEFINE
The number on the bottom of the fraction, which tells us how many equal parts the whole has been cut into.

Mixed number
DEFINE
A number that combines whole numbers and fractions.

Simplify
DEFINE
Express a fraction using the smallest numbers possible, without changing the amount that the fraction represents.

KeepChangeFlip
DEFINE
A method that helps us remember how to change a division problem into a multiplication problem.

Fraction
DEFINE

Reading Material
Download as PDF Download PDF View as Separate PageWHAT IS DIVIDING FRACTIONS BY FRACTIONS?Previously, you learned how to divide a whole number by a fraction and a fraction by a whole number. Today you learn that the same processes you already know can help you divide fractions by fractions.
To better understand dividing fractions by fractions…
WHAT IS DIVIDING FRACTIONS BY FRACTIONS?. Previously, you learned how to divide a whole number by a fraction and a fraction by a whole number. Today you learn that the same processes you already know can help you divide fractions by fractions. To better understand dividing fractions by fractions…LET’S BREAK IT DOWN!
Divide 5 by [ggfrac]1/4[/ggfrac]
Imagine we serve 5 quesadillas at a party, and each one of them is cut into fourths. To find out how many pieces we have in total, we need to divide 5 by [ggfrac]1/4[/ggfrac]. If we draw 5 circles and split each of them into 4 equal parts, we have a total of 20 pieces. We can rewrite division problems as multiplication problems: We keep the first number the same, we change the operation to multiplication, and we flip fraction so [ggfrac]1/4[/ggfrac] becomes 4. Then 5 ÷ [ggfrac]1/4[/ggfrac] is the same as 5 x 4, which is 20! Now you try: Solve 3 ÷ [ggfrac]1/5[/ggfrac].
Divide 5 by [ggfrac]1/4[/ggfrac] Imagine we serve 5 quesadillas at a party, and each one of them is cut into fourths. To find out how many pieces we have in total, we need to divide 5 by [ggfrac]1/4[/ggfrac]. If we draw 5 circles and split each of them into 4 equal parts, we have a total of 20 pieces. We can rewrite division problems as multiplication problems: We keep the first number the same, we change the operation to multiplication, and we flip fraction so [ggfrac]1/4[/ggfrac] becomes 4. Then 5 ÷ [ggfrac]1/4[/ggfrac] is the same as 5 x 4, which is 20! Now you try: Solve 3 ÷ [ggfrac]1/5[/ggfrac].Divide 3 by [ggfrac]3/4[/ggfrac]
To solve 3 ÷ [ggfrac]3/4[/ggfrac], we follow the keepchangeflip method again to turn the division problem into a multiplication problem: We keep the 3, we change the operation to multiplication, and we flip the fraction. Then 3 ÷ [ggfrac]3/4[/ggfrac] becomes 3 x [ggfrac]4/3[/ggfrac] which is equal to [ggfrac]12/3[/ggfrac], which simplifies to 4! Now you try: Solve 4 ÷ [ggfrac]2/3[/ggfrac].
Divide 3 by [ggfrac]3/4[/ggfrac] To solve 3 ÷ [ggfrac]3/4[/ggfrac], we follow the keepchangeflip method again to turn the division problem into a multiplication problem: We keep the 3, we change the operation to multiplication, and we flip the fraction. Then 3 ÷ [ggfrac]3/4[/ggfrac] becomes 3 x [ggfrac]4/3[/ggfrac] which is equal to [ggfrac]12/3[/ggfrac], which simplifies to 4! Now you try: Solve 4 ÷ [ggfrac]2/3[/ggfrac].Dividing Fractions by Fractions
A community garden is [ggfrac]3/4[/ggfrac] of an acre, and we want to divide it into sections that are [ggfrac]1/8[/ggfrac] of an acre. How many sections are there? We can solve this by dividing [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/8[/ggfrac]. Even though we are dividing a fraction by a fraction, we can still use the keepchangeflip method. We keep [ggfrac]3/4[/ggfrac], we change the operation to multiplication, and we flip [ggfrac]1/8[/ggfrac] so that it becomes [ggfrac]8/1[/ggfrac]. Now the problem is the same as [ggfrac]3/4[/ggfrac] x [ggfrac]8/1[/ggfrac]. Remember that when we multiply fractions, we multiply the numerator by the numerator, and the denominator by the denominator. 3×8=24 and 4×1=4. So our answer is [ggfrac]24/4[/ggfrac], which simplifies to 6! The garden has 6 sections. Now you try: Solve [ggfrac]4/6[/ggfrac] ÷ [ggfrac]1/3[/ggfrac].
Dividing Fractions by Fractions A community garden is [ggfrac]3/4[/ggfrac] of an acre, and we want to divide it into sections that are [ggfrac]1/8[/ggfrac] of an acre. How many sections are there? We can solve this by dividing [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/8[/ggfrac]. Even though we are dividing a fraction by a fraction, we can still use the keepchangeflip method. We keep [ggfrac]3/4[/ggfrac], we change the operation to multiplication, and we flip [ggfrac]1/8[/ggfrac] so that it becomes [ggfrac]8/1[/ggfrac]. Now the problem is the same as [ggfrac]3/4[/ggfrac] x [ggfrac]8/1[/ggfrac]. Remember that when we multiply fractions, we multiply the numerator by the numerator, and the denominator by the denominator. 3×8=24 and 4×1=4. So our answer is [ggfrac]24/4[/ggfrac], which simplifies to 6! The garden has 6 sections. Now you try: Solve [ggfrac]4/6[/ggfrac] ÷ [ggfrac]1/3[/ggfrac].Quotients That Are Mixed Numbers
You have [ggfrac]5/6[/ggfrac] yard of fabric that you want to make shirts out of. Each shirt needs [ggfrac]2/5[/ggfrac] yard of fabric. How many shirts can you make? We can find out by solving [ggfrac]5/6[/ggfrac] ÷ [ggfrac]2/5[/ggfrac]. If we use the keepchangeflip method, we can turn this into the multiplication problem [ggfrac]5/6[/ggfrac] x [ggfrac]5/2[/ggfrac], which is equal to [ggfrac]25/12[/ggfrac]. Since 25 is greater than 12, we know that this fraction is greater than one whole. 25 divided by 12 is 2 with a remainder of 1, so this is equal to 2[ggfrac]1/12[/ggfrac]. That means we can make 2 shirts, and we will have a little bit of fabric left over. Now you try: Solve [ggfrac]8/5[/ggfrac] ÷ [ggfrac]1/3[/ggfrac].
Quotients That Are Mixed Numbers You have [ggfrac]5/6[/ggfrac] yard of fabric that you want to make shirts out of. Each shirt needs [ggfrac]2/5[/ggfrac] yard of fabric. How many shirts can you make? We can find out by solving [ggfrac]5/6[/ggfrac] ÷ [ggfrac]2/5[/ggfrac]. If we use the keepchangeflip method, we can turn this into the multiplication problem [ggfrac]5/6[/ggfrac] x [ggfrac]5/2[/ggfrac], which is equal to [ggfrac]25/12[/ggfrac]. Since 25 is greater than 12, we know that this fraction is greater than one whole. 25 divided by 12 is 2 with a remainder of 1, so this is equal to 2[ggfrac]1/12[/ggfrac]. That means we can make 2 shirts, and we will have a little bit of fabric left over. Now you try: Solve [ggfrac]8/5[/ggfrac] ÷ [ggfrac]1/3[/ggfrac].Dividing Using Mixed Numbers
If you run a total of 2[ggfrac]3/4[/ggfrac] miles on a track and each lap is [ggfrac]11/12[/ggfrac] mile, how many laps did you run? We can find out by solving 2[ggfrac]3/4[/ggfrac] ÷ [ggfrac]11/12[/ggfrac]. First, we need to convert the mixed number into a fraction that is greater than 1. We want to express the whole number 2 using the denominator 4. 2 = [ggfrac]4/4[/ggfrac] + [ggfrac]4/4[/ggfrac] = [ggfrac]8/4[/ggfrac], plus [ggfrac]3/4[/ggfrac] equals [ggfrac]11/4[/ggfrac] in total. Now we have [ggfrac]11/4[/ggfrac] ÷ [ggfrac]11/12[/ggfrac] and we can use the keepchangeflip method to rewrite the problem as [ggfrac]11/4[/ggfrac] x [ggfrac]12/11[/ggfrac]. This is equal to [ggfrac]132/44[/ggfrac], which simplifies to 3! You ran 3 laps on the track. Now you try: Solve 1[ggfrac]2/6[/ggfrac] ÷ [ggfrac]2/3[/ggfrac].
Dividing Using Mixed Numbers If you run a total of 2[ggfrac]3/4[/ggfrac] miles on a track and each lap is [ggfrac]11/12[/ggfrac] mile, how many laps did you run? We can find out by solving 2[ggfrac]3/4[/ggfrac] ÷ [ggfrac]11/12[/ggfrac]. First, we need to convert the mixed number into a fraction that is greater than 1. We want to express the whole number 2 using the denominator 4. 2 = [ggfrac]4/4[/ggfrac] + [ggfrac]4/4[/ggfrac] = [ggfrac]8/4[/ggfrac], plus [ggfrac]3/4[/ggfrac] equals [ggfrac]11/4[/ggfrac] in total. Now we have [ggfrac]11/4[/ggfrac] ÷ [ggfrac]11/12[/ggfrac] and we can use the keepchangeflip method to rewrite the problem as [ggfrac]11/4[/ggfrac] x [ggfrac]12/11[/ggfrac]. This is equal to [ggfrac]132/44[/ggfrac], which simplifies to 3! You ran 3 laps on the track. Now you try: Solve 1[ggfrac]2/6[/ggfrac] ÷ [ggfrac]2/3[/ggfrac]. 
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Solve [ggfrac]1/5[/ggfrac] ÷ [ggfrac]3/5[/ggfrac].
Solve [ggfrac]1/3[/ggfrac] ÷ 2[ggfrac]1/2[/ggfrac].
Solve 1[ggfrac]1/4[/ggfrac] ÷ 2[ggfrac]2/3[/ggfrac].
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