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calendar_month Last update: 2025-12-22

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Multiplying and dividing fractions booklet

calendar_month 2025-12-22

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Unit 1: Numbers
Unit 2: Geometry and measure
Unit 3: Statistics and probability

🎯 In this topic you will

Multiply a unit fraction by a whole number
Divide a unit fraction by a whole number

🧠 Key Words

repeated addition
unit fraction

Show Definitions

repeated addition: Adding the same number multiple times; a way to understand multiplication.
unit fraction: A fraction with a numerator of 1, representing one part of a whole divided into equal pieces.

Understanding the Pictures

L ook at the pictures. They show how multiplying a unit fraction by a whole number can make a complete whole.

Sharing a Fraction

I magine you have one half of an apple and you share it with four friends. This helps you think about dividing a unit fraction among several people.

Using Calculations

T his idea helps you write a calculation to show how much of the fruit each person gets when a unit fraction is shared.

Learning the Goal

I n this section you will learn how to multiply and divide unit fractions, using simple examples to build understanding.

📘 Worked example

a. Calculate $ \dfrac{1}{5} \times 4 $.
Draw a diagram to show your answer.

You can use different types of diagram.

The bar model shows a whole divided into fifths with four-fifths shaded.

The number line shows $ \dfrac{1}{5} \times 4 $ as repeated addition of $ \dfrac{1}{5} $.

Answer:

a. $ \dfrac{1}{5} \times 4 = \dfrac{4}{5} $

Four groups of $ \dfrac{1}{5} $ make a total of $ \dfrac{4}{5} $.

The bar model shows four fifths shaded, and the number line shows four jumps of size $ \dfrac{1}{5} $.

❓ EXERCISES

1. Calculate $\dfrac{1}{4} \times 3$. Draw a diagram to show your answer.

👀 Show answer

$\dfrac{1}{4} \times 3 = \dfrac{3}{4}$

2. Calculate $\dfrac{1}{5} \times 6$. Draw a diagram to show your answer.

👀 Show answer

$\dfrac{1}{5} \times 6 = \dfrac{6}{5} = 1\dfrac{1}{5}$

3. Amy, Kiki and Magda work out $\dfrac{1}{6} \times 4$. Here are their methods.

👀 Show answer

$\dfrac{1}{6} \times 4 = \dfrac{4}{6} = \dfrac{2}{3}$ Each method shows four jumps or four blocks of size $\dfrac{1}{6}$.

4. Draw a diagram to help you calculate $\dfrac{1}{3} \times 4$.

👀 Show answer

$\dfrac{1}{3} \times 4 = \dfrac{4}{3} = 1\dfrac{1}{3}$

5. Arun multiplies a unit fraction by a whole number. He writes:

$\dfrac{1}{5} \times 5 = \dfrac{5}{25}$

Explain what Arun has done wrong.

👀 Show answer

Arun multiplied both the numerator and the denominator by $5$. Correct multiplication is: $\dfrac{1}{5} \times 5 = \dfrac{5}{5} = 1$.

6. Draw diagrams to help you calculate

a. $\dfrac{1}{5} \div 2$

b. $\dfrac{1}{6} \div 3$

c. $\dfrac{1}{4} \div 5$

Check your answers with your partner.

👀 Show answer

a. $\dfrac{1}{5} \div 2 = \dfrac{1}{10}$

b. $\dfrac{1}{6} \div 3 = \dfrac{1}{18}$

c. $\dfrac{1}{4} \div 5 = \dfrac{1}{20}$

7. Zara has $\dfrac{1}{3}$ of a bottle of fruit juice. She divides it equally between two glasses. What fraction of the bottle is in each glass?

👀 Show answer

$\dfrac{1}{3} \div 2 = \dfrac{1}{6}$ Each glass contains one sixth of the bottle.

🧠 Think like a Mathematician

Look at these pairs of calculations.

$\dfrac{1}{2} \times 7 = \dfrac{7}{2}$     $7 \div 2 = \dfrac{7}{2}$
$\dfrac{1}{2} \times 6 = \dfrac{6}{2}$     $6 \div 2 = \dfrac{6}{2}$
$\dfrac{1}{2} \times 5 = \dfrac{5}{2}$     $5 \div 2 = \dfrac{5}{2}$

Write the next three rows of the pattern.

What do you notice about multiplying by $\dfrac{1}{2}$ and dividing by 2?

What happens if you multiply by $\dfrac{1}{3}$ and divide by 3?

You will show that you are generalising when you explain the pattern and find examples that satisfy the pattern.

You will show that you are convincing when you explain the relationship between multiplying by $\dfrac{1}{3}$ and dividing by 3.

Show Answers

Next three rows of the pattern:
$\dfrac{1}{2} \times 4 = \dfrac{4}{2}$     $4 \div 2 = \dfrac{4}{2}$
$\dfrac{1}{2} \times 3 = \dfrac{3}{2}$     $3 \div 2 = \dfrac{3}{2}$
$\dfrac{1}{2} \times 2 = \dfrac{2}{2}$     $2 \div 2 = \dfrac{2}{2}$
What do you notice?
Multiplying any number by $\dfrac{1}{2}$ gives exactly the same result as dividing that number by 2.
What happens with $\dfrac{1}{3}$ and dividing by 3?
Multiplying a number by $\dfrac{1}{3}$ gives the same result as dividing the number by 3. In both cases you are finding one third of the number.
Generalising the pattern:
Multiplying by the unit fraction $\dfrac{1}{n}$ is equivalent to dividing by $n$.

📘 What we've learned

We learned how to multiply a unit fraction by a whole number by thinking of repeated addition: $k \times \frac{1}{n}=\frac{k}{n}$.
We learned to simplify or rewrite answers when needed (for example, converting $\frac{k}{n}$ to a mixed number if $k \ge n$).
We learned how to divide a unit fraction by a whole number by making the denominator $k$ times larger: $\frac{1}{n}\div k=\frac{1}{n\cdot k}$.
We learned that dividing by a whole number makes the fraction smaller, so $\frac{1}{n}\div k$ is always less than $\frac{1}{n}$ when $k>1$.

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