# Multiplying and Dividing Fractions

> Multiply tops and bottoms, cancel first to keep numbers small, and divide by flipping the second fraction.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/multiplying-and-dividing-fractions/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Half of half a chocolate bar
Ben has half a chocolate bar. His little sister begs for some, so he gives her half of what he has.

How much of the whole bar did she get? Half of a half. If you draw it, you see it is one quarter of the bar. In maths, "of" means multiply:

$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Then Ben wonders how many quarter-bar pieces fit in 3 whole bars. Each bar has 4 quarters, so 12 pieces. That is division: $3 \div \frac{1}{4} = 12$.

**In short:** To multiply fractions, multiply the tops and multiply the bottoms. To divide, keep the first fraction, change $\div$ to $\times$ and flip the second one, so $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2$.

## Before you start

- Simplifying fractions by dividing top and bottom by a common factor.
- Changing mixed numbers to improper fractions: $2\frac{1}{3} = \frac{7}{3}$.
- A whole number can be written as a fraction over 1: $5 = \frac{5}{1}$.

## Multiplying fractions

### Key idea: Top times top, bottom times bottom
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

$\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$

You do **not** need a common denominator to multiply. That is only for adding and subtracting.

## Cancel first

Before multiplying, divide any top number and any bottom number by a common factor. This keeps the numbers small.

### Worked example: Cross cancelling
$\frac{7}{12} \times \frac{4}{21}$

7 and 21 share 7: they become 1 and 3. 4 and 12 share 4: they become 1 and 3.

$\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$

### Worked example: Another
$\frac{5}{9} \times \frac{18}{35}$

5 and 35 share 5: they become 1 and 7. 18 and 9 share 9: they become 2 and 1.

$\frac{1}{1} \times \frac{2}{7} = \frac{2}{7}$

## Mixed numbers: change them first

### Watch out: Do not multiply the parts separately
$1\frac{1}{2} \times 2\frac{2}{3}$ is **not** $2\frac{2}{6}$. Change both to improper fractions first.

$\frac{3}{2} \times \frac{8}{3}$. Cancel the 3s and turn 8 and 2 into 4 and 1. The answer is $4$.

### Worked example: Mixed times proper
$3\frac{1}{8} \times \frac{4}{5} = \frac{25}{8} \times \frac{4}{5}$

Cancel 25 and 5 to 5 and 1. Cancel 4 and 8 to 1 and 2.

$\frac{5}{2} \times \frac{1}{1} = \frac{5}{2} = 2\frac{1}{2}$

## Dividing fractions

Dividing asks "how many of these fit into that?". There is a neat rule.

### Key idea: Keep, change, flip
1. **Keep** the first fraction.
2. **Change** $\div$ to $\times$.
3. **Flip** the second fraction upside down.

$$\frac{3}{4} \div \frac{9}{8} = \frac{3}{4} \times \frac{8}{9} = \frac{2}{3}$$

The flipped fraction is called the **reciprocal**. The reciprocal of $\frac{9}{8}$ is $\frac{8}{9}$, and the reciprocal of 6 is $\frac{1}{6}$.

### Worked example: Dividing mixed numbers
$1\frac{2}{3} \div \frac{5}{9} = \frac{5}{3} \times \frac{9}{5} = \frac{9}{3} = 3$

$6\frac{2}{3} \div \frac{1}{6} = \frac{20}{3} \times \frac{6}{1} = 40$

### Watch out: Flip the second one only
In $\frac{3}{4} \div \frac{9}{8}$, flip $\frac{9}{8}$, not $\frac{3}{4}$. Also change mixed numbers to improper fractions **before** flipping.

## How many parts in a mixed number?

"How many thirds are in $3\frac{2}{3}$?" Change to an improper fraction: $\frac{11}{3}$. There are 11 thirds.

This is the same as $3\frac{2}{3} \div \frac{1}{3} = 11$.

### Challenge: Selective level: longer divisions
$2\frac{3}{4} \div 2\frac{1}{10} = \frac{11}{4} \div \frac{21}{10} = \frac{11}{4} \times \frac{10}{21} = \frac{110}{84} = \frac{55}{42} = 1\frac{13}{42}$

$\frac{3}{8}$ of a number is 27. What is the number? One eighth is $27 \div 3 = 9$, so the number is $9 \times 8 = 72$. That is the same as $27 \div \frac{3}{8} = 27 \times \frac{8}{3} = 72$.

A ribbon $4\frac{1}{2}$ m long is cut into pieces $\frac{3}{8}$ m long. Number of pieces: $\frac{9}{2} \times \frac{8}{3} = 12$.

**Mind map**

```text
Multiply and divide fractions
  Multiply
    Top x top
    Bottom x bottom
    No common denominator needed
  Cancel first
    Any top with any bottom
  Mixed numbers
    Make improper first
  Divide
    Keep, change, flip
    Reciprocal
  How many parts
    Thirds in $3\frac{2}{3}$ is 11
```

## Quick summary

- Multiply fractions: top times top, bottom times bottom.
- Cancel common factors between any top and any bottom first.
- Change mixed numbers to improper fractions before multiplying or dividing.
- Divide: keep the first, change $\div$ to $\times$, flip the second.
- "How many thirds in $3\frac{2}{3}$?" means change it to an improper fraction and read the top.

## Quiz

Try these questions from the chapter quiz.

1. $\frac{3}{4} \times \frac{2}{5} = $
   - A. $\frac{5}{9}$
   - B. $\frac{3}{10}$
   - C. $\frac{15}{8}$
   - D. $\frac{6}{9}$
   - Answer: B. $\frac{3}{10}$. $\frac{6}{20} = \frac{3}{10}$.

2. What is the reciprocal of $\frac{9}{8}$?
   - A. $\frac{8}{9}$
   - B. $\frac{9}{8}$
   - C. $1\frac{1}{8}$
   - D. $\frac{1}{8}$
   - Answer: A. $\frac{8}{9}$. Flip it upside down: $\frac{8}{9}$.

3. How many thirds are in $3\frac{2}{3}$?
   - A. 5
   - B. 9
   - C. 11
   - D. 6
   - Answer: C. 11. $3\frac{2}{3} = \frac{11}{3}$, which is 11 thirds.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/multiplying-and-dividing-fractions/#quiz