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

Multiplying Fractions by Whole Numbers and Mixed Numbers

B2.6 A1About 45 minutes
Learning goals
  • I can multiply a fraction by a whole number, a fraction or a mixed number.
  • I can simplify before I multiply by dividing out common factors.
  • I can use an area model to explain why I cannot just multiply the whole parts and the fraction parts.
  • I can estimate a product and solve problems like scaling a recipe.

Words to know

improper fraction
A fraction with a numerator greater than or equal to its denominator, like 5/2.
common factor
A number that divides two numbers evenly. 3 is a common factor of 3 and 9.
simplify before multiplying
Divide a numerator and a denominator by a common factor before you multiply, to keep the numbers small.
area model
A rectangle whose side lengths are the factors and whose area is the product.

Remember from Grade 7

In Grade 7 you multiplied fractions by fractions: multiply the numerators, multiply the denominators, then simplify. You also saw that "of" means multiply: 2/3 of 9/10 = 2/3 × 9/10. In Grade 8 you will multiply fractions by whole numbers and mixed numbers in real problems, and simplify before you multiply.

Let's start with maple cookies

A cookie recipe uses 2/3 cup of maple syrup. You want to make 2 1/2 batches for a school bake sale. How much maple syrup do you need?

2 1/2 × 2/3 = 5/2 × 2/3 = 10/6 = 1 2/3 cups.

Estimate to check: 2 batches need 2 × 2/3 = 1 1/3 cups and 3 batches need 2 cups. 1 2/3 is between them. ✓

Round cookies cooling on a wire rack beside a glass measuring cup and a small jug of maple syrup.
A recipe that uses 2/3 cup of maple syrup per batch needs 2 1/2 × 2/3 = 1 2/3 cups for 2 1/2 batches. AI-generated image
2 1/2 batches × 2/3 cup of syrup per batchbatch 12/3batch 22/31/2 batch1/3=1 cup2/3 cup5 thirds= 1 2/3cups
Each batch needs 2 thirds of a cup, and half a batch needs 1 third. That is 5 thirds in all: 2 1/2 × 2/3 = 5/3 = 1 2/3 cups.

The method

  1. Write whole numbers over 1 (6 = 6/1) and change mixed numbers to improper fractions (2 1/2 = 5/2).
  2. Simplify before multiplying: if a numerator and a denominator share a factor, divide both by it.
  3. Multiply the numerators and multiply the denominators.
  4. Write the answer in lowest terms, as a mixed number if it is improper.

Simplifying first example: 10/3 × 3/5. The 3s divide to 1, and 10 and 5 divide by 5 to give 2 and 1. So the product is 2/1 × 1/1 = 2.

21/211/32 × 1 = 21/22/31/62 + 2/3 + 1/2 + 1/6 = 3 1/3
An area model for 2 1/2 × 1 1/3 (not to scale). The four parts are 2 × 1 = 2, 1/2 × 1 = 1/2, 2 × 1/3 = 2/3 and 1/2 × 1/3 = 1/6. Together they make 3 1/3.

Watch out!

You cannot just multiply the whole numbers and the fractions separately. 2 1/2 × 1 1/3 is not 2 × 1 + 1/2 × 1/3 = 2 1/6. The area model shows two more parts (1/2 and 2/3) that would be left out. The correct product is 5/2 × 4/3 = 20/6 = 3 1/3.

Worked example

Example 1: Whole number × fraction

Find 6 × 3/4.

  1. 6/1 × 3/4. Simplify: 6 and 4 share a factor of 2, giving 3/1 × 3/2.
  2. 3 × 3 = 9 and 1 × 2 = 2, so 9/2.
  3. 9/2 = 4 1/2.

4 1/2

Worked example

Example 2: Mixed number × fraction

Find 3 1/3 × 3/5.

  1. 3 1/3 = 10/3, so 10/3 × 3/5.
  2. Simplify: the 3s divide to 1, and 10 and 5 divide by 5 to give 2 and 1.
  3. 2/1 × 1/1 = 2.

2

Worked example

Example 3: Mixed × mixed

Find 2 1/4 × 1 1/3.

  1. 9/4 × 4/3.
  2. Simplify: the 4s divide to 1, and 9 and 3 divide by 3 to give 3 and 1.
  3. 3/1 × 1/1 = 3.

3

Worked example

Example 4: In context

A park trail is 4 1/2 km long. Jade has walked 2/3 of it. How far has she walked?

  1. 2/3 of 4 1/2 = 2/3 × 9/2.
  2. Simplify: 2s divide to 1, 9 and 3 divide by 3 to give 3 and 1.
  3. 1/1 × 3/1 = 3.

3 km

Bigger or smaller?

Multiplying by a number less than 1 makes the answer smaller (3/4 of 8 is 6). Multiplying by a number greater than 1 makes it bigger (1 1/2 × 8 = 12). Use this as a quick check.

Hands-on activity

Scale a recipe

You need: a family recipe (or one from a cookbook or package), paper, a pencil.

  1. Choose 4 ingredients with fraction or mixed-number amounts.
  2. Rewrite the recipe for 1 1/2 batches. Multiply each amount by 1 1/2.
  3. Rewrite it for 2/3 of a batch. Multiply each amount by 2/3.
  4. Estimate first each time, then compute, then compare.
  5. If you decide to bake, a grown-up handles the oven and anything hot.

Model answers: 3/4 cup milk × 1 1/2 = 3/4 × 3/2 = 9/8 = 1 1/8 cups. 2 1/4 cups flour × 2/3 = 9/4 × 2/3 = 18/12 = 1 1/2 cups. 1/2 tsp salt × 2/3 = 1/3 tsp.

Math talk: "Why did every amount get smaller for 2/3 of a batch?"

Just for fun

Why is multiplying by 1/2 the worst way to share your fries?

Every time you do it, there are fewer fries left.

Your turn

1

5 × 2/3 = ?

2

3/8 × 16 = ?

3

1 1/2 × 4/5 = ?

4

2 2/3 × 3/4 = ?

5

1 3/4 × 2 2/5 = ?

6

A movie is 1 1/2 hours long. You have watched 2/3 of it. How long have you watched?

7

A rectangle is 3 1/2 m by 2 1/4 m. What is its area?

More practice: warm-up

1

4 × 1/2 = ?

2

1/3 × 9 = ?

3

2/5 × 10 = ?

4

1/2 × 1/2 = ?

5

3 × 1/4 = ?

More practice: on your own

1

2 1/2 × 6 = ?

2

5/6 × 1 1/5 = ?

3

3 1/3 × 2 1/2 = ?

4

A jug holds 2 1/4 L. How much do 3 jugs hold?

5

Estimate, then find 4 1/4 × 1 7/8.

Challenge

1

Find 1 1/2 × 1 1/3 × 1 1/4.

2

Kai's bedroom floor is 3 3/4 m by 3 1/5 m. Carpet costs $24 per square metre. How much will the carpet cost?

3

Find (−2/3) × 1 1/2.

Check your understanding

Quiz

1.

What is 4 × 2/3?

2.

What is 1 1/2 × 2/3?

3.

Why is 2 1/2 × 1 1/2 not equal to 2 1/4?

4.

What is 3/5 of 20?

5.

Which product is less than 5?

6.

What is 2 1/3 × 3?

For parents and teachers

Big idea: Ontario's Grade 8 expectation B2.6 asks students to multiply and divide fractions by fractions, as well as by whole numbers and mixed numbers, in various contexts. This lesson covers multiplication; the next covers division.

Common difficulties:

  • Multiplying wholes and fractions separately (2 1/2 × 1 1/2 = 2 1/4). The area model shows why this misses parts.
  • Multiplying both parts of a fraction by a whole number (4 × 2/3 = 8/12).
  • Forgetting to change mixed numbers to improper fractions.

How to help: Scale a real recipe together. Ask for an estimate first, and ask "Should the answer be bigger or smaller than the original amount?"