Grade by Grade
B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Multiplying Whole Numbers by Proper Fractions

B2.9 A1About 40 minutes
Learning goals
  • I can explain that 3/4 × 12 means 3/4 of 12.
  • I can multiply a whole number by a proper fraction using a model or a strategy.
  • I can write an answer as a whole number, a fraction or a mixed number.

Words to know

proper fraction
A fraction less than 1, where the numerator is smaller than the denominator, like 3/4 or 2/5.
of
In math, "3/4 of 12" means 3/4 × 12.
product
The answer when you multiply.
mixed number
A whole number and a fraction together, like 3 1/3.

Remember from Grade 5

In Grade 5 you multiplied whole numbers by unit fractions: 6 × 1/3 = 2, because one third of 6 is 2. Now you'll multiply by any proper fraction, like 3/4, 2/3 or 5/8.

Let's start with apples

You picked 12 apples at an orchard in the fall. You want to bake with 3/4 of them. How many is that?

  • Split the 12 apples into 4 equal groups (the denominator). Each group has 3.
  • Take 3 of those groups (the numerator). 3 groups of 3 is 9.

So 3/4 of 12 = 9, and 3/4 × 12 = 12 × 3/4 = 9.

🍎🍎🍎🍎🍎🍎🍎🍎🍎🍎🍎🍎
12 apples in 4 equal groups of 3. One group is 1/4 of 12. Three groups are 3/4 of 12: 9 apples.

Two strategies

Strategy 1: divide, then multiply. Divide the whole number by the denominator, then multiply by the numerator.

3/4 of 12: 12 ÷ 4 = 3, and 3 × 3 = 9.

Strategy 2: multiply, then divide. Multiply the whole number by the numerator to count the pieces. Then turn the pieces into wholes.

12 × 3/4 = 36/4 (that's 36 fourths) = 36 ÷ 4 = 9.

Strategy 2 is helpful when the denominator doesn't divide evenly into the whole number:

5 × 2/3 = 10/3 = 3 1/3.

Smaller answer: multiplying a whole number by a proper fraction always gives an answer less than the whole number, because you are taking part of it.

Worked example

Example 1: Divide, then multiply

There are 40 students in the Grade 6 classes. 3/8 of them walk to school. How many walk?

  1. 3/8 of 40 means 3/8 × 40.
  2. Divide by the denominator: 40 ÷ 8 = 5. That's 1/8.
  3. Multiply by the numerator: 5 × 3 = 15. That's 3/8.

15 students walk to school.

Worked example

Example 2: The answer is a mixed number

5 × 2/3 = ?

  1. Think of 5 groups of 2/3: 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 10/3.
  2. Or multiply the numerator: 5 × 2 = 10 thirds = 10/3.
  3. 10/3 = 9/3 + 1/3 = 3 1/3.

5 × 2/3 = 3 1/3

5 × 2/3 = 10 thirds = 3 1/32/32/32/32/32/3put the 10 thirds together1 whole1 whole1 whole1/3= 3 1/3
Five groups of 2/3 make 10 thirds. Nine of them fill 3 wholes, and 1 third is left: 3 1/3.
Worked example

Example 3: Bannock for a crowd

Many First Nations, Métis and Inuit families make bannock, a kind of bread. Kayla's family recipe uses 3/4 cup of milk for one batch. She is making 3 batches for a community feast. How much milk does she need?

  1. 3 batches × 3/4 cup = 3 × 3/4.
  2. 3 × 3 = 9 fourths = 9/4.
  3. 9/4 = 8/4 + 1/4 = 2 1/4.

Kayla needs 2 1/4 cups of milk.

Watch out

Only multiply the numerator by the whole number. The denominator tells the size of the pieces, and that doesn't change. 3 × 3/4 = 9/4, not 9/12.

Just for fun

Baking with 3/4 of your 12 apples leaves 3 apples. That's 1/4 of the apples, but 100% of the crunchy snack.

Hands-on activity

Fraction of a handful

You need: 24 small items such as dry pasta pieces, buttons or crackers, and a plate. (Keep small items away from little brothers and sisters, since they can be a choking hazard.)

  1. Count out 24 items. Find 1/3 of them by making 3 equal groups. How many in one group? (8)
  2. Now find 2/3 of 24. (16)
  3. Find 3/4, 5/6 and 3/8 of 24 by making equal groups. (18, 20, 9)
  4. Write each one as a multiplication, like 2/3 × 24 = 16.
  5. Now try 5 × 3/4: make 5 groups of 3 quarter-crackers (or draw them). How many whole crackers is that?

Math talk: "Which number did you divide by? Which number did you multiply by?"

Your turn

1

2/3 of 15 = ?

2

3/5 of 20 = ?

3

4 × 3/4 = ?

4

6 × 2/5 = ?

5

24 × 5/6 = ?

6

7 × 3/8 = ?

7

3/10 of 50 = ?

More practice: warm-up

1

1/4 of 20 = ?

2

1/3 of 12 = ?

3

2 × 1/5 = ?

4

3/4 of 8 = ?

5

3 × 1/3 = ?

More practice: on your own

1

5/8 of 32 = ?

2

9 × 2/3 = ?

3

10 × 3/4 = ?

4

A recipe uses 2/3 cup of oats. You make 4 batches. How many cups of oats?

5

There are 36 cars in a parking lot. 7/12 of them are white. How many white cars?

Challenge

1

Which is greater: 4/5 of 45 or 5/6 of 42?

2

2/3 of a number is 14. What is the number?

3

A school has 240 students. 5/8 walk or bike. The rest come by bus or car. How many come by bus or car?

Check your understanding

Quiz

1.

What is 2/3 of 18?

2.

What is 5 × 3/4?

3.

Which means the same as 3/5 × 30?

4.

What is 3/8 of 64?

5.

When you multiply 12 by a proper fraction, the answer is…

6.

What is 4 × 2/7?

For parents and teachers

Big idea: B2.9 asks Grade 6 students to multiply whole numbers by proper fractions, using appropriate tools and strategies. Grade 5 covered unit fractions only (like 1/3). The key meaning: 3/4 × 12 is "3/4 of 12". Students may divide by the denominator first, or multiply by the numerator first.

Common difficulties:

  • Multiplying the denominator too (3 × 3/4 = 9/12).
  • Expecting a bigger answer, because multiplication "always makes bigger".
  • Leaving an improper fraction when a mixed number is easier to understand in context (9/4 cups = 2 1/4 cups).

How to help: Use real objects and recipes. Ask "What is 2/3 of the 18 grapes on your plate?" Bannock is used here as one example of food many First Nations, Métis and Inuit families make today; if your family has its own recipes or traditions, they make great math contexts too.