# Improper Fractions And Mixed Numbers

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.

## What are improper fractions and mixed numbers?

• Improper fractions and proper fractions are types of fractions.

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

E.g.

$\frac{1}{2} \quad \quad \frac{3}{4} \quad \quad \frac{17}{20}$

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”.

E.g.

$\frac{7}{2} \quad \quad \frac{13}{4} \quad \quad \frac{14}{5}$

• A mixed number has a whole number part and a fractional part

E.g.

$1\frac{3}{5}$

This means 1 whole and 3 fifths.

We can change improper fractions to mixed numbers and vice versa.

1 whole is split into 5 equal pieces so 1 whole is equal to 5 fifths.

Altogether there are 8 fifths or 1 whole and 3 fifths.

$\frac{8}{5}=1\frac{3}{5}$

## How to convert improper fractions and mixed numbers

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

1. Work out how many times the denominator divides into the numerator.
2. Work out the remainder.
3. Write the mixed number.

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

1. Multiply the whole number by the denominator.
3. Write the improper fraction.

## Improper fraction and mixed number examples

### Example 1: change an improper fraction to a mixed number

Write the following improper fraction as a mixed number:

$\frac{7}{3}$

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

The denominator of the fraction is 3 and the numerator of the fraction is 7.

3 divides into 7 twice.  This gives the whole number 2.

2Work out the remainder.

The remainder is 1. This gives the numerator for the fractional part.
The denominator stays the same.

$\frac{7}{3}=\frac{2\times3+1}{3}=2\frac{1}{3}$

3Write the mixed number.

$\frac{7}{3}=2\frac{1}{3}$

### Example 2:  change an improper fraction to a mixed number

Write the following improper fraction as a mixed number:

$\frac{17}{5}$

The denominator of the fraction is 5 and the numerator of the fraction is 17.

5 divides into 17 three times.  This gives the whole number 3.

The remainder is 2. This gives the numerator for the fractional part.
The denominator stays the same.

$\frac{17}{5}=\frac{3\times5+2}{5}=3\frac{2}{5}$

$\frac{17}{5}=3\frac{2}{5}$

### Example 3: change an improper fraction to a mixed number

Write the following improper fraction as a mixed number:

$\frac{17}{7}$

The denominator of the fraction is 7 and the numerator of the fraction is 17.

7 divides into 17 three times.  This gives the whole number 2.

The remainder is 3. This gives the numerator for the fractional part.  The denominator stays the same.

$\frac{7}{3}=\frac{2\times7+3}{7}=2\frac{3}{7}$

$\frac{17}{7}=2\frac{3}{7}$

### Example 4: change a mixed number to an improper fraction

Write the following mixed number as an improper fraction:

$2\frac{4}{5}$

The whole number part of the mixed number is 2 and the denominator is 5.

$2\times5=10$

So

$2=\frac{10}{5}$

The numerator is 4.

$10+4=14$

The new numerator for the improper fraction is 14.

$2\frac{4}{5}=\frac{2\times5+4}{5}=\frac{10+4}{5}$

$2\frac{4}{5}=\frac{14}{5}$

### Example 5: change a mixed number to an improper fraction

Write the following mixed number as an improper fraction:

$3\frac{1}{6}$

The whole number part of the mixed number is 3 and the denominator is 6.

$3\times6=18$

So

$3=\frac{18}{6}$

The numerator is 1.

$18+1=19$

The new numerator for the improper fraction is 19.

$3\frac{1}{6}=\frac{3\times6+1}{6}=\frac{18+1}{6}$

$3\frac{1}{6}=\frac{19}{6}$

### Example 6: change a mixed number to an improper fraction

Write the following mixed number as an improper fraction:

$4\frac{3}{7}$

The whole number part of the mixed number is 4 and the denominator is 7.

$4\times7=28$

So

$4=\frac{28}{7}$

The numerator is 3.

$28+3=31$

The new numerator for the improper fraction is 31.

$4\frac{3}{7}=\frac{4\times7+3}{7}=\frac{28+3}{7}$

$4\frac{3}{7}=\frac{31}{7}$

### Common misconceptions

• The denominator (the bottom number) stays the same

If the improper fraction has a denominator of 3, then so will the mixed number and vice versa.

• 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.

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

### Practice improper fraction and mixed number questions

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}

### Improper fraction and mixed number GCSE questions

1.  Write this improper fraction as a mixed number

\frac{25}{8}

(1 mark)

\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}

(1 mark)

\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

(2 marks)

\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)

## Learning checklist

You have now learned how to:

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

## Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.