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Addition and subtraction of whole numbers

Multiplying and dividing of whole numbers

This topic is relevant for:

Here we will learn about improper fractions and mixed numbers including how to recognise improper fractions and mixed numbers and how to convert between them.

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

**Proper fractions**and**improper fractions**are types of fractions.

A **proper fraction** is a fraction where the **numerator **(top number) is **smaller **than the **denominator **(bottom number).

Here are some examples of proper fractions:

\[\frac{1}{2} \quad \quad \frac{3}{4} \quad \quad \frac{17}{20}\]

An **improper **fraction is a fraction where the **numerator **(top number) is **larger **than the **denominator **(bottom number). They are sometimes called “top-heavy fractions”.

Here are some examples of improper fractions:

\[\frac{7}{2} \quad \quad \frac{13}{4} \quad \quad \frac{14}{5}\]

- A
**mixed number**has a**whole number part**and a**fractional part**.

Here is an example of a mixed number, it means

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

We can change improper fractions to mixed numbers and vice versa. To do this we know how many equal parts there are and how many of them make a whole.

There is

Altogether there are

\[\frac{8}{5}=1\frac{3}{5}\]

In order to change an improper fraction to a mixed number:

- Work out how many times the denominator divides into the numerator.
- Work out the remainder.
- Write the mixed number.

In order to change a mixed number to an improper fraction:

- Multiply the whole number by the denominator.
- Add on the numerator.
- Write the improper fraction.

Get your free improper fraction and mixed number worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free improper fraction and mixed number worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONWrite the following improper fraction as a mixed number:

\[\frac{7}{3}\]

**Work out how many times the denominator divides into the numerator**.

The denominator of the fraction is

2**Work out the remainder**.

The remainder is

\[\frac{7}{3}=\frac{2\times3+1}{3}=2\frac{1}{3}\]

3**Write the mixed number**.

\[\frac{7}{3}=2\frac{1}{3}\]

Write the following improper fraction as a mixed number:

\[\frac{17}{5}\]

**Work out how many times the denominator divides into the numerator**.

The denominator of the fraction is

**Work out the remainder**.

The remainder is

\[\frac{17}{5}=\frac{3\times5+2}{5}=3\frac{2}{5}\]

**Write the mixed number**.

\[\frac{17}{5}=3\frac{2}{5}\]

Write the following improper fraction as a mixed number:

\[\frac{17}{7}\]

**Work out how many times the denominator divides into the numerator**.

The denominator of the fraction is

**Work out the remainder**.

The remainder is

\[\frac{7}{3}=\frac{2\times7+3}{7}=2\frac{3}{7}\]

**Write the mixed number**.

\[\frac{17}{7}=2\frac{3}{7}\]

Write the following mixed number as an improper fraction:

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

**Multiply the whole number by the denominator**.

The whole number part of the mixed number is

\[2\times5=10\]

So

\[2=\frac{10}{5}\]

**Add on the numerator**.

The numerator is

\[10+4=14\]

The new numerator for the improper fraction is

\[2\frac{4}{5}=\frac{2\times5+4}{5}=\frac{10+4}{5}\]

**Write the improper fraction**.

\[2\frac{4}{5}=\frac{14}{5}\]

Write the following mixed number as an improper fraction:

\[3\frac{1}{6}\]

**Multiply the whole number by the denominator**.

The whole number part of the mixed number is

\[3\times6=18\]

So

\[3=\frac{18}{6}\]

**Add on the numerator**.

The numerator is

\[18+1=19\]

The new numerator for the improper fraction is

\[3\frac{1}{6}=\frac{3\times6+1}{6}=\frac{18+1}{6}\]

**Write the improper fraction**.

\[3\frac{1}{6}=\frac{19}{6}\]

Write the following mixed number as an improper fraction:

\[4\frac{3}{7}\]

**Multiply the whole number by the denominator**.

The whole number part of the mixed number is

\[4\times7=28\]

So

\[4=\frac{28}{7}\]

**Add on the numerator**.

The numerator is

\[28+3=31\]

The new numerator for the improper fraction is

\[4\frac{3}{7}=\frac{4\times7+3}{7}=\frac{28+3}{7}\]

**Write the improper fraction**.

\[4\frac{3}{7}=\frac{31}{7}\]

**The denominator (the bottom number) stays the same**

If the improper fraction has a denominator of

**Take care when using a calculator and mixed numbers**

When putting a mixed number into a calculator, you must use the “shift” button and then the fraction button so that you can input the mixed number properly.

**Sometimes the fraction will need to be written in its simplest terms**

An improper fraction may be written as a mixed number but the fraction part of the mixed number still needs to be written in its simplest terms.

\[\frac{22}{4}=\frac{5\times4+2}{4}=5\frac{2}{4}=5\frac{1}{2}\]

1. Write the following improper fraction as a mixed number:

\frac{5}{4}

1\frac{1}{4}

\frac{4}{5}

1\frac{1}{5}

1\frac{3}{4}

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

2. Write the following improper fraction as a mixed number:

\frac{8}{3}

2\frac{2}{3}

\frac{3}{8}

1\frac{2}{3}

2\frac{1}{3}

\frac{8}{3}=\frac{2\times3+2}{3}=2\frac{2}{3}

3. Write the following improper fraction as a mixed number:

\frac{23}{5}

4\frac{3}{5}

\frac{5}{23}

2\frac{3}{5}

3\frac{4}{5}

\frac{23}{5}=\frac{4\times5+3}{5}=4\frac{3}{5}

4. Write the following mixed number as an improper fraction:

2\frac{3}{5}

\frac{13}{5}

\frac{5}{23}

2\frac{1}{5}

3\frac{2}{5}

2\frac{3}{5}=\frac{2\times5+3}{5}=\frac{10+3}{5}=\frac{13}{5}

5. Write the following mixed number as an improper fraction:

5\frac{2}{7}

\frac{37}{7}

\frac{52}{7}

\frac{25}{7}

\frac{34}{7}

5\frac{2}{7}=\frac{5\times7+2}{7}=\frac{35+2}{7}=\frac{37}{7}

6. Write the following mixed number as an improper fraction:

8\frac{1}{4}

\frac{33}{4}

\frac{81}{4}

\frac{33}{8}

\frac{35}{4}

8\frac{1}{4}=\frac{8\times4+1}{4}=\frac{32+1}{4}=\frac{33}{4}

1. Write this improper fraction as a mixed number

\frac{25}{8}

**(1 mark)**

Show answer

\frac{25}{8}=\frac{3\times8+1}{8}=3\frac{1}{8}

3\frac{1}{8}

**(1)**

2. Work out

\frac{4}{9} + \frac{7}{9}

Circle your answer

\frac{11}{18} \quad \quad \frac{11}{81} \quad \quad 1\frac{4}{9} \quad \quad 1\frac{2}{9}

**(1 mark)**

Show answer

\frac{4}{9}+\frac{7}{9}=\frac{4+7}{9}=\frac{11}{9}=\frac{1\times9+2}{9}=1\frac{2}{9}

1\frac{2}{9}

**(1)**

3. Work out

\frac{3}{7}\times11

Give your answer as a mixed number

**(2 marks)**

Show answer

\frac{3}{7}\times11=\frac{33}{7}

**(1)**

\frac{33}{7}=\frac{4\times 7+5}{7}=4\frac{5}{7}

4\frac{5}{7}

**(1)**

You have now learned how to:

- Convert an improper fraction to a mixed number
- Convert a mixed number to an improper fraction

- Equivalent fractions
- Ordering fractions
- Adding and subtracting fractions
- Multiplying and dividing fractions

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