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Simplifying fractions

Converting between mixed and improper fractionsMultiplying and dividing whole numbers

Multiplying fractionsThis topic is relevant for:

Here we will learn about **dividing fractions** including how to divide fractions, divide fractions by whole numbers, and divide mixed fractions.

There are also dividing fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Dividing fractions** is where we find the** reciprocal of (flip) the second fraction**, **change the divide sign** to a multiply and then** multiply the fractions** together.

E.g.

There are

We need to find how many halves there are in

Let’s illustrate this with a diagram:

There are

So,

\[6 \div \frac{1}{2}=12\]

We notice that this is the same as

So to **divide two fractions we flip the second fraction and multiply them**.

\[\begin{array}{l}
6 \div \frac{1}{2} \\\\
6 \times \frac{2}{1} \\\\
=12 \text { people }
\end{array}\]

The **reciprocal **of a number is a number which, when multiplied by the number itself, equals

E.g.

\[\frac{2}{1} \times \frac{1}{2}= 1\]

So,

\[\frac{1}{2}\]

is the reciprocal of

\[\frac{2}{1}\]

In order to divide two or more fractions:

**Flip the second fraction (find its reciprocal)****Change the divide sign to multiplication****Multiply the fractions together****Simplify if possible**

E.g. **Divide two proper fractions**

\[\frac{1}{2} \div \frac{1}{3}\]

Flip the second fraction:

\[\frac{1}{2} \div \frac{3}{1}\]

Multiply:

\[\frac{1}{2} \times \frac{3}{1}=\frac{3}{2}\]

Simplify if possible:

\[\frac{3}{2} \]

E.g. **Divide a fraction by a whole number**

\[\frac{1}{2} \div 4\]

Put the whole number over

\[\frac{1}{2} \div \frac{4}{1}\]

Flip the second fraction:

\[\frac{1}{2} \div \frac{1}{4}\]

Multiply:

\[\frac{1}{2} \times \frac{1}{4}=\frac{1}{8}\]

Simplify if possible:

\[\frac{1}{8} \]

E.g. **Divide two mixed number fractions**

\[1 \frac{1}{2} \div 2 \frac{1}{4}\]

Change the mixed fractions into improper fractions:

\[\frac{3}{2} \div \frac{9}{4}\]

Flip the second fraction:

\[\frac{3}{2} \div \frac{4}{9}\]

Multiply:

\[\frac{3}{2} \times \frac{4}{9}=\frac{12}{18}\]

Simplify if possible:

\[\frac{2}{3} \]

Get your free Dividing fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free Dividing fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEDivide the fractions below:

\[\frac{1}{7} \div \frac{1}{2}\]

**Flip the second fraction (find its reciprocal)**

\[\frac{1}{7} \div \frac{2}{1}\]

Turn the second fraction upside down. The numerator becomes the denominator and vice versa.

2**Change the divide sign to multiplication**

\[\frac{1}{7} \times \frac{2}{1}\]

3**Multiply the fractions together**

\[\frac{1}{7} \times \frac{2}{1}\]

*Remember: To multiply two fractions together multiply the numerators together and the denominators together.*

\[\frac{1 \times 2}{7 \times 1}=\frac{2}{7}\]

4**Simplify if possible**

The fraction cannot be simplified so we are left with

\[\frac{2}{7} \]

Divide the fractions below:

\[\frac{1}{5} \div \frac{1}{2} \div \frac{2}{3}\]

**Flip the second fraction (find its reciprocal)**

In this case we have three fractions so flip the third fraction as well (find its reciprocal).

\[\frac{1}{5} \div \frac{2}{1} \div \frac{3}{2}\]

Turn the second and third fractions upside down. The numerator becomes the denominator and vice versa.

**Change the divide sign to multiplication**

\[\frac{1}{5} \times \frac{2}{1} \times \frac{3}{2}\]

**Multiply the fractions together**

\[\frac{1}{5} \times \frac{2}{1} \times \frac{3}{2}\]

*Remember: To multiply two fractions together multiply the numerators together and the denominators together.*

\[\frac{1 \times 2 \times 3}{5 \times 1 \times 2}=\frac{6}{10}\]

**Simplify if possible**

\[\frac{6}{10}\]

can be simplified to

\[\frac{6 \div 2}{10 \div 2}=\frac{3}{5}\]

In order to divide fractions by whole numbers:

**Put the whole number over**1 **Flip the second fraction (find its reciprocal)****Change the divide sign to multiplication****Multiply the fractions together****Simplify if possible**

Divide the fractions below:

\[\frac{1}{3} \div \ 4\]

**Put the whole number over 1**

\[\frac{1}{3} \div \frac{4}{1}\]

**Flip the second fraction (find its reciprocal)**

\[\frac{1}{3} \div \frac{1}{4}\]

Turn the second fraction upside down. The numerator becomes the denominator and vice versa.

**Change the divide sign to multiplication**

\[\frac{1}{3} \times \frac{1}{4}\]

**Multiply the fractions together**

\[\frac{1}{3} \times \frac{1}{4}\]

*Remember: To multiply two fractions together multiply the numerators together and the denominators together.*

\[\frac{1 \times 1}{3 \times 4}=\frac{1}{12}\]

**Simplify if possible**

The fraction cannot be simplified so we are left with

\[\frac{1}{12}\]

There are

**First we need to create an equation to model the problem**

Since

We need to divide \frac{1}{4} into

As an equation:

\[12 \div \frac{1}{4}\]

Now we are ready to perform the calculation:

**Put the whole number over 1**

\[\frac{12}{1} \div \frac{1}{4}\]

**Flip the second fraction (find its reciprocal)**

\[\frac{12}{1} \div \frac{4}{1}\]

Turn the second fraction upside down. The numerator becomes the denominator and vice versa.

**Change the divide sign to multiplication**

\[\frac{12}{1} \times \frac{4}{1}\]

**Multiply the fractions together**

\[\frac{12}{1} \times \frac{4}{1}\]

\[\frac{12 \times 4}{1 \times 1}=\frac{48}{1}\]

**Simplify if possible**

We are left with

In order to divide mixed fractions:

**Change mixed fraction(s) to improper fraction(s)****Flip the second fraction (find its reciprocal)****Change the divide sign to multiplication****Multiply the fractions together****Simplify and convert back to a mixed number if possible**

Divide the fractions below:

\[1 \frac{1}{3} \div 2 \frac{1}{2}\]

**Change mixed fractions to improper fractions**

*Remember: To convert a mixed number into an improper fraction multiply the denominator by the whole number and then add it to the numerator.*

\[1 \frac{+1}{\times3}=\frac{4}{3}\]

\[ 2 \frac{+1}{\times2}=\frac{5}{2}\]

**Flip the second fraction (find its reciprocal)**

\[\frac{4}{3} \div \frac{2}{5}\]

Turn the second fraction upside down. The numerator becomes the denominator and vice versa.

**Change the divide sign to multiplication**

\[\frac{4}{3} \times \frac{2}{5}\]

**Multiply the fractions together**

\[\frac{4}{3} \times \frac{2}{5}\]

\[\frac{4 \times 2}{3 \times 5}=\frac{8}{15}\]

**Simplify if possible**

The fraction cannot be simplified so we are left with

\[ \frac{8}{15} \]

A rectangle has an area of 1 \frac{1}{4} m^{2} and a width of \frac{1}{5} m . What is the length of the rectangle?

**First we need to create an equation to model the problem**

If we divide the width into the total area we will be able to calculate the length.

\[l=1 \frac{1}{4} \div \frac{1}{5}\]

Now we are ready to perform the calculation:

**Change mixed fractions to improper fractions**

*Remember: To convert a mixed number into an improper fraction multiply the denominator by the whole number and then add it to the numerator.*

\[1 \frac{+1}{\times4}=\frac{5}{4}\]

**Flip the second fraction (find its reciprocal)**

\[\frac{5}{4} \div \frac{5}{1}\]

Turn the second fraction upside down. The numerator becomes the denominator and vice versa.

**Change the divide sign to multiplication**

\[\frac{5}{4} \times \frac{5}{1}\]

**Multiply the fractions together**

\[\frac{5}{4} \times \frac{5}{1}\]

\[\frac{5 \times 5}{4 \times 1}=\frac{25}{4}\]

**Simplify if possible**

The fraction cannot be simplified so we are left with a length of

\[\frac{25}{4}m= 6 \frac{1}{4} \mathrm{m} \]

**Thinking that dividing by \frac{1}{2} is the same as halving a number**

E.g.

\[4 \div \frac{1}{2} \neq 2\]

**Forgetting to flip and change the sign**

A common error is to forget to flip and change the sign to multiply and instead just divide the numerators together and the denominators together.

E.g.

\[\frac{9}{10} \div \frac{3}{5} \neq \frac{3}{2}\]

**Flipping the first fraction instead of the second fraction**

A common error is to flip the first fraction instead of the second

E.g.

\[\frac{9}{10} \div \frac{3}{5}\]

\[\frac{10}{9} \div \frac{3}{5}\]

This is incorrect.

**Cross multiplying instead of multiplying numerators together and denominators together**

E.g.

\[\frac{2}{3} \times \frac{1}{4} \neq \frac{8}{3}\]

Dividing fractions is part of our series of lessons to support revision on fractions. You may find it helpful to start with the main fractions lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

1. \frac{4}{11} \div \frac{1}{7}

\frac{4}{77}

\frac{17}{77}

\frac{28}{11}

\frac{81}{11}

Flip the second fraction and change divide to multiply:

\begin{aligned} \frac{4}{11} \div \frac{1}{7} &= \frac{4}{11} \times \frac{7}{1}\\\\ &=\frac{28}{11} \end{aligned}

2. \frac{2}{3} \div \frac{6}{10}

\frac{10}{9}

\frac{12}{18}

\frac{21}{9}

\frac{9}{10}

Flip the second fraction and change divide to multiply:

\begin{aligned} \frac{2}{3} \div \frac{6}{10} &= \frac{2}{3} \times \frac{10}{6}\\\\ &= \frac{20}{18} \end{aligned}

Simplify the fraction: \frac{20}{18} = \frac{10}{9}

3. \frac{2}{9} \div \ 5

\frac{0.4}{1.8}

\frac{10}{9}

\frac{10}{45}

\frac{2}{45}

First we need to rewrite 5 as \frac{5}{1} .

We need to calculate \frac{2}{9} \div \frac{5}{1} .

Flip the second fraction and change divide to multiply:

\begin{aligned} \frac{2}{9} \div \frac{5}{1} &= \frac{2}{9} \times \frac{1}{5}\\\\ &=\frac{2}{45} \end{aligned}

4. A seamstress has 15 yards of fabric on the roll. She needs \frac{1}{3} yard of fabric to sew each garment. How many garments can she make?

5

45

18

12

We need to calculate 15 \div \frac{1}{3} .

First we rewrite 15 as \frac{15}{1} and then flip the second fraction and change the divide to multiply.

\begin{aligned} \frac{15}{1} \div \frac{1}{3} &= \frac{15}{1} \times \frac{3}{1}\\\\ &= \frac{45}{1} \end{aligned}

\frac{45}{1} is 45 so she can make 45 garments.

5. 3 \frac{2}{3} \div 1 \frac{1}{5}

3 \frac{1}{18}

6 \frac{1}{3}

3 \frac{10}{3}

4 \frac{6}{15}

First we need to write the mixed numbers as improper fractions.

3 \frac{2}{3} = \frac{11}{3}\\

1 \frac{1}{5} = \frac{6}{5}

We need to calculate \frac{11}{3} \div \frac{6}{5} .

Flip the second fraction and change the divide to multiply:

\begin{aligned} \frac{11}{3} \div \frac{6}{5} &= \frac{11}{3} \times \frac{5}{6}\\\\ &= \frac{55}{18} \end{aligned}

Convert the fraction back to a mixed number: \frac{55}{18} = 3 \frac{1}{18}

6. There are 5 \frac{1}{3} pizzas at a party. Each person ate 1 \frac{1}{3} pizzas. How many people ate pizzas?

6

5

4

3

We need to calculate 5 \frac{1}{3} \div 1 \frac{1}{3} .

First we write the improper fractions as mixed numbers:

5 \frac{1}{3} = \frac{16}{3}

1 \frac{1}{3} = \frac{4}{3}

We need to calculate \frac{16}{3} \div \frac{4}{3} .

Flip the second fraction and change the divide to multiply:

\begin{aligned} \frac{16}{3} \div \frac{4}{3} &= \frac{16}{3} \times \frac{3}{4}\\\\ &= \frac{48}{12} \end{aligned}

Simplify the fraction: \frac{48}{12}=4

1. Work out \frac{3}{5} \div \frac{1}{7}

**(3 marks)**

Show answer

\frac{7}{1}

Evidence of flipping second fraction

**(1)**

Evidence of multiplication (multiplication sign seen)

**(1)**

\frac{21}{5}

**(1)**

2. Work out 1 \frac{1}{4} \div 2 \frac{2}{5} . Express your answer in the simplest form.

**(4 marks)**

Show answer

\frac{5}{4} or \frac{12}{5} seen (converts to improper fraction)

**(1)**

Evidence of flipping the second fraction

**(1)**

Evidence of multiplication

**(1)**

\frac{25}{48}

**(1)**

3. (a) There are 24 pizzas eaten at a party. Each friend eats \frac{2}{3} of a pizza. How many friends were at the party?

**(4 marks)**

Show answer

24 \div \frac{2}{3}

**(1)**

Evidence of finding the reciprocal of the second fraction \frac{3}{2} AND changing the divide to a multiply.

**(1)**

36 friends

**(1)**

You have now learned how to:

- Use the multiplication operation, including formal written methods, applied to proper and improper fractions and mixed numbers

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