B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Dividing Fractions by Fractions

B2.8 A1About 45 minutes

Learning goals

I can explain dividing fractions as "how many of this fit into that?"
I can divide fractions using a common denominator.
I can divide fractions by multiplying by the reciprocal.
I can check a fraction division with multiplication.

Words to know

quotient
The answer to a division.

reciprocal
The fraction flipped upside down. The reciprocal of 2/3 is 3/2. A number times its reciprocal is 1.

divisor
The number you divide by. In 3/4 ÷ 1/8, the divisor is 1/8.

dividend
The number being divided. In 3/4 ÷ 1/8, the dividend is 3/4.

Remember from Grade 6

In Grade 6 you divided a whole number by a fraction by asking "How many fit?" For 8 ÷ 2/3 you counted how many 2/3s fit into 8 (12 of them). Now you will divide a fraction by a fraction, using the same idea.

Let's start with juice

You have 3/4 L of apple juice. Each small cup holds 1/8 L. How many cups can you fill?

You are asking how many 1/8s fit into 3/4. That is 3/4 ÷ 1/8. The picture shows that 3/4 is the same as 6 eighths, so you can fill 6 cups.

A strip from 0 to 1 cut into eighths. The part from 0 to 3/4 is six eighths, so 3/4 ÷ 1/8 = 6.

Strategy 1: Common denominator

Write both fractions with the same denominator. Then just divide the numerators, because you are counting same-size pieces.

3/4 ÷ 1/8 = 6/8 ÷ 1/8 = 6 ÷ 1 = 6
3/4 ÷ 2/3 = 9/12 ÷ 8/12 = 9 ÷ 8 = 9/8 = 1 1/8

Strategy 2: Multiply by the reciprocal

To divide by a fraction, multiply by its reciprocal (flip the divisor):

3/4 ÷ 2/3 = 3/4 × 3/2 = 9/8

Why does it work? Think of dividing by 1/3. There are 3 thirds in every whole, so dividing by 1/3 is the same as multiplying by 3. Dividing by 2/3 gives half as many groups as dividing by 1/3, so it is the same as multiplying by 3/2.

Only flip the divisor (the second fraction). Change mixed numbers to improper fractions first.

How many 2/3s fit in 3/4? One full group of 8 twelfths fits, plus 1 more twelfth, which is 1/8 of a group. So 3/4 ÷ 2/3 = 1 1/8.

Worked example

Example 1: Common denominator

Find 2/3 ÷ 1/6.

LCD of 3 and 6 is 6. 2/3 = 4/6.
4/6 ÷ 1/6 = 4 ÷ 1 = 4.
Check: 4 × 1/6 = 4/6 = 2/3. ✓

2/3 ÷ 1/6 = 4. Four sixths fit into two thirds.

Worked example

Example 2: Reciprocal, answer less than 1

Find 1/2 ÷ 2/3.

Flip the divisor: the reciprocal of 2/3 is 3/2.
1/2 × 3/2 = 3/4.
Does it make sense? 2/3 is bigger than 1/2, so a whole 2/3 doesn't fit. Only part of it fits: 3/4 of it.
Check: 3/4 × 2/3 = 6/12 = 1/2. ✓

1/2 ÷ 2/3 = 3/4

Worked example

Example 3: Mixed numbers in context

You have 2 1/4 cups of flour. Each batch of pancakes needs 3/4 cup. How many batches can you make?

How many 3/4s fit into 2 1/4? 2 1/4 = 9/4.
9/4 ÷ 3/4: same denominators, so 9 ÷ 3 = 3.
Check: 3 × 3/4 = 9/4 = 2 1/4. ✓

3 batches

Think like a mathematician

Predict the size before you calculate. If the divisor is smaller than the dividend, the answer is more than 1. If the divisor is bigger, the answer is less than 1. And dividing by a fraction less than 1 gives an answer bigger than the number you started with, because many small pieces fit.

Hands-on activity

Measuring-cup division

You need: measuring cups (1 cup, 1/2, 1/3 and 1/4 cup), rice or dry beans, a bowl, and a tray to catch spills.

Fill the 1/2 cup with rice. How many 1/4 cups can you fill from it? Write 1/2 ÷ 1/4 = ?
Fill the 1 cup. How many 1/3 cups fit? Write 1 ÷ 1/3 = ?
Fill the 1/2 cup and three 1/4 cups of rice into a bowl (that's 1 1/4 cups). How many 1/4 cups is that? Write 1 1/4 ÷ 1/4 = ?
Fill the 1/2 cup. Pour it into the 1/3 cup. It fills once, with some rice left over. The leftover fills half of the 1/3 cup. Write 1/2 ÷ 1/3 = ?

What to look for: 1/2 ÷ 1/4 = 2. 1 ÷ 1/3 = 3. 1 1/4 ÷ 1/4 = 5. 1/2 ÷ 1/3 = 1 1/2 (check: 3/6 ÷ 2/6 = 3/2).

Math talk: "Why is 1/2 ÷ 1/3 more than 1?"

Just for fun

Dividing by 1/2 is the same as multiplying by 2. So if someone offers you 10 cookies divided by one-half, say yes. That's 20 cookies!

Your turn

1/2 ÷ 1/4 = ?

3/4 ÷ 1/8 = ?

2/3 ÷ 1/6 = ?

5/6 ÷ 1/3 = ?

3/5 ÷ 3/10 = ?

1/2 ÷ 2/3 = ?

2 1/4 ÷ 3/4 = ?

More practice: warm-up

1 ÷ 1/5 = ?

1/2 ÷ 1/6 = ?

4/5 ÷ 2/5 = ?

What is the reciprocal of 3/7?

3/4 ÷ 1/4 = ?

More practice: on your own

7/8 ÷ 1/4 = ?

5/6 ÷ 5/9 = ?

A piece of ribbon is 2/3 m long. How many 1/6 m bows can you cut from it?

A jug holds 3 1/2 L of water. How many 1/4 L glasses can you fill?

4/9 ÷ 2/3 = ?

Challenge

2 1/2 ÷ 1 1/4 = ?

Ella says, "Dividing always makes a number smaller, so 3/4 ÷ 1/2 must be less than 3/4." Is she right? Explain and find the answer.

A rectangle has an area of 3/4 m² and a width of 1/2 m. How long is it?

Check your understanding

Quiz

What is 3/4 ÷ 1/4?

What is the reciprocal of 5/8?

What is 2/3 ÷ 3/4?

How many 1/3 cup scoops are in 2 cups?

Which division has an answer greater than 1?

Which is the best way to check 3/5 ÷ 1/10 = 6?

For parents and teachers

Big idea: The Ontario curriculum (B2.8) asks Grade 7 students to multiply and divide fractions by fractions, using tools in various contexts. Students should understand division as "how many of the divisor fit into the dividend" before relying on the "multiply by the reciprocal" shortcut.

Common difficulties:

Flipping the wrong fraction (the dividend instead of the divisor), or flipping both.
Expecting a smaller answer: dividing by a fraction less than 1 usually gives a bigger answer.
Mixed numbers: forgetting to change them to improper fractions first.

How to help: Do the measuring-cup activity together. When your child uses the reciprocal rule, ask: "Can you show me why with a common denominator?" This lesson builds A1 skills: reasoning, proving with a model, and checking with the inverse operation.