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]**

**Divide 3 by [ggfrac]3/4[/ggfrac]**

To solve 3 ÷ [ggfrac]3/4[/ggfrac], we follow the keep-change-flip 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 keep-change-flip 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**

**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 keep-change-flip 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 keep-change-flip 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**

**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 keep-change-flip 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 keep-change-flip 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**

**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 keep-change-flip 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 keep-change-flip 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 FRACTIONS BY FRACTIONS VOCABULARY**

**DIVIDING FRACTIONS BY FRACTIONS VOCABULARY**

**Fraction**

**A number that expresses equal parts of a whole.**

**Numerator**

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

**Denominator**

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

**Mixed number**

**A number that combines whole numbers and fractions.**

**Simplify**

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

**Keep-Change-Flip**

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

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 one-half-sized pieces there are in three-fourths. [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 three-fourth-sized 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. ## DIVIDING FRACTIONS BY FRACTIONS DISCUSSION QUESTIONS

### How can you turn [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/2[/ggfrac] into a multiplication problem?

### To turn a division problem into a multiplication problem, could you instead flip the first fraction?

### Is [ggfrac]3/4[/ggfrac] ÷ [ggfrac]1/2[/ggfrac] the same as [ggfrac]1/2[/ggfrac] ÷ [ggfrac]3/4[/ggfrac]? Explain why or why not.

### What is the first thing you should do to divide [ggfrac]1/2[/ggfrac] ÷ 1[ggfrac]3/5[/ggfrac]?

### If you solve [ggfrac]1/2[/ggfrac] ÷ 1[ggfrac]3/5[/ggfrac], should the answer be less than or greater than [ggfrac]1/2[/ggfrac]?

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