B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Dividing with Fractions, Whole Numbers and Mixed Numbers

B2.6 A1About 45 minutes

Learning goals

I can divide a whole number by a fraction and a fraction by a whole number.
I can divide mixed numbers using common denominators or by multiplying by the reciprocal.
I can explain what a quotient means: how many groups, or how much in each group.
I can check a quotient by multiplying, and decide what to do with a remainder in a real problem.

Words to know

reciprocal
The number you multiply by to get 1. The reciprocal of 3/4 is 4/3; the reciprocal of 5 is 1/5; the reciprocal of 1 2/3 (= 5/3) is 3/5.

quotient
The answer to a division.

divisor
The number you divide by.

common denominator
A shared denominator. With a common denominator, divide the numerators: 9/4 ÷ 3/4 = 9 ÷ 3 = 3.

Remember from Grade 7

In Grade 7 you divided fractions by fractions, using common denominators or by multiplying by the reciprocal. In Grade 8 you will also divide whole numbers by fractions, fractions by whole numbers and mixed numbers by mixed numbers, and decide what the answer means in a real situation.

Let's start with a ribbon

You have 3 m of ribbon for hanging decorations. Each decoration needs 3/4 m. How many decorations can you hang?

The question is "how many 3/4s fit into 3?" That is 3 ÷ 3/4. The picture shows that 4 pieces fit exactly.

Three wholes, each cut into quarters. Groups of three quarters (3/4) are shaded. Exactly 4 groups fit, so 3 ÷ 3/4 = 4.

Two meanings of division

How many groups? 3 ÷ 3/4 asks how many pieces of size 3/4 fit into 3. Answer: 4.
How much in each group? 3/4 ÷ 3 asks how much each person gets if 3/4 of a pizza is shared by 3. Answer: 1/4 each.

Two methods

Multiply by the reciprocal. a ÷ b = a × (reciprocal of b). Change mixed numbers to improper fractions first.

3 ÷ 3/4 = 3 × 4/3 = 12/3 = 4.
3/4 ÷ 3 = 3/4 × 1/3 = 3/12 = 1/4.

Common denominators. Write both numbers with the same denominator, then divide the numerators.

3 3/4 ÷ 1 1/4 = 15/4 ÷ 5/4 = 15 ÷ 5 = 3.

Both methods always give the same answer. Choose the one that looks easier.

"How much in each group?" Sharing 3/4 of a pizza among 3 friends gives each one quarter: 3/4 ÷ 3 = 1/4.

Worked example

Example 1: Whole number ÷ fraction

Find 5 ÷ 2/3.

Multiply by the reciprocal: 5 × 3/2 = 15/2.
15/2 = 7 1/2.
Meaning: 7 full portions of 2/3 fit into 5, plus half of another portion.

7 1/2

Worked example

Example 2: Fraction ÷ whole number

Find 4/5 ÷ 2.

Multiply by the reciprocal of 2, which is 1/2: 4/5 × 1/2 = 4/10.
4/10 = 2/5. (Makes sense: half of 4 fifths is 2 fifths.)

2/5

Worked example

Example 3: Mixed ÷ mixed with common denominators

Find 3 3/4 ÷ 1 1/4.

3 3/4 = 15/4 and 1 1/4 = 5/4. They already share the denominator 4.
15 ÷ 5 = 3.

Worked example

Example 4: Mixed ÷ mixed with the reciprocal

Find 2 1/2 ÷ 1 2/3.

5/2 ÷ 5/3 = 5/2 × 3/5.
Simplify the 5s: 1/2 × 3/1 = 3/2 = 1 1/2.
Check: 1 1/2 × 1 2/3 = 3/2 × 5/3 = 15/6 = 2 1/2. ✓

1 1/2

Worked example

Example 5: Bags of trail mix

You have 6 3/4 cups of trail mix. How many 3/4-cup bags can you fill?

27/4 ÷ 3/4. Same denominator, so divide the numerators: 27 ÷ 3 = 9.

9 bags

Check by multiplying

Quotient × divisor = dividend. If 5 ÷ 2/3 = 7 1/2, then 7 1/2 × 2/3 = 15/2 × 2/3 = 30/6 = 5. ✓

Also check the size: dividing by a number less than 1 gives a bigger answer (5 ÷ 2/3 is more than 5), and dividing by a number greater than 1 gives a smaller answer.

Hands-on activity

How many scoops?

You need: measuring cups (1 cup, 1/2 cup, 1/3 cup, 1/4 cup), a bowl of dry rice or oats, an empty bowl, a tray for spills.

Measure 1 1/2 cups into the empty bowl using the 1 cup and the 1/2 cup. Predict how many 1/4 cups it holds. Scoop it out with the 1/4 cup and count.
Measure 2 cups. Predict, then count, how many 1/3 cups it holds.
Measure 2 1/2 cups. Predict, then count, how many 1/2 cups it holds.
Write a division sentence for each, and check one by multiplying.

Model answers: 1. 1 1/2 ÷ 1/4 = 6. 2. 2 ÷ 1/3 = 6. 3. 2 1/2 ÷ 1/2 = 5. Check: 6 × 1/4 = 6/4 = 1 1/2. ✓

Just for fun

How many half-pizzas can you get from 3 pizzas? 3 ÷ 1/2 = 6. Dividing by a fraction less than 1 is the only time math hands you more slices than you started with (they are just smaller).

Your turn

4 ÷ 1/3 = ?

6 ÷ 3/4 = ?

2/3 ÷ 4 = ?

3 1/2 ÷ 1/2 = ?

4 1/2 ÷ 1 1/2 = ?

1 3/4 ÷ 2 1/3 = ?

A 10 m rope is cut into pieces that are each 1 1/4 m long. How many pieces are there?

More practice: warm-up

2 ÷ 1/2 = ?

3 ÷ 1/4 = ?

1/2 ÷ 2 = ?

3/4 ÷ 1/4 = ?

5 ÷ 1/5 = ?

More practice: on your own

9 ÷ 1 1/2 = ?

5/6 ÷ 5 = ?

2 2/3 ÷ 1 1/3 = ?

3 3/5 ÷ 2 1/4 = ?

7 1/2 kg of birdseed is shared equally among 6 bird feeders. How much goes in each?

Challenge

A cyclist rides 12 1/2 km in 5/6 of an hour. What is her speed in kilometres per hour?

Find 1/2 ÷ 1/4 ÷ 1/8.

How many 2/3 L glasses can you fill from a 5 L jug? How much is left over?

Check your understanding

Quiz

What is 3 ÷ 1/4?

What is 2/5 ÷ 2?

What is 2 1/2 ÷ 1/2?

What is the reciprocal of 1 2/3?

What is 3 1/3 ÷ 1 2/3?

Three friends share 3/4 of a pizza equally. How much does each get?

For parents and teachers

Big idea: This lesson finishes Ontario's Grade 8 expectation B2.6: multiply and divide fractions by fractions, as well as by whole numbers and mixed numbers, in various contexts. Students should understand why dividing by a fraction less than 1 gives a bigger answer, not just follow "flip and multiply".

Common difficulties:

Flipping the wrong fraction (the first instead of the divisor).
Forgetting to change mixed numbers to improper fractions before finding the reciprocal (the reciprocal of 1 2/3 is not 1 3/2).
Remainders in context: 7 1/2 glasses means 7 full glasses.

How to help: Do the measuring-cup activity. Ask "How many of these fit?" questions in the kitchen, and ask your child to check answers by multiplying.