GCSE Maths Number

Fractions, Decimals and Percentages

# Fractions, Decimals And Percentages

Here we will learn about fractions, decimals and percentages, including what they are, how to calculate with them and to solve problems involving them.

There are also fractions, decimals and percentages 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 fractions, decimals and percentages?

Fractions, decimals and percentages are different ways of representing a proportion of the same amount.

There is equivalence between fractions, decimals and percentages.

E.g.
\frac{43}{100}=0.43=43\%

### What are fractions, decimals and percentages? ### What are fractions?

Fractions are a way of writing equal parts of one whole.

They have a numerator (top number) and a denominator (bottom number).
The denominator shows how many equal parts the whole has been divided into.
The numerator shows how many of the equal parts we have.

E.g.
This shape has 9 equal parts and 4 of them are shaded.
This represents four ninths: \frac{4}{9}

### What are decimals?

Decimals are a way of writing numbers that are not whole.

Decimal numbers can be recognised as they have a decimal point.
A decimal place is a position after the decimal point.

E.g.
0.37 has two decimal places.

There is a 3 in the tenths place and 7 in the hundredths place.

E.g.
This shows the fraction \frac{7}{10}

\frac{7}{10} can also be written as 0.7

### What are percentages?

Percentages are numbers which are expressed as parts of 100 .

Percent means “number of parts per hundred” and the symbol we use for this is the percent sign (%).

E.g. 43\%

There are 100 equal parts and 43 of them are shaded.

## How to use fractions, decimals and percentages

There are various ways of using fractions, decimals and percentages.

For examples, practice questions and worksheets on each one follow the links to the step by step guides below:

## Comparing fractions, decimals and percentages

In order to compare fractions, decimals and percentages you need to be able to convert between them, including:

• Converting fractions to decimals
• Converting decimals to fractions
• Converting fractions to percentages
• Converting percentages to fractions
• Converting decimals to percentages
• Converting percentages to decimals
• Converting recurring decimals to fractions

Step-by-step guide: Converting fractions, decimals and percentages

## Fractions

E.g.

\begin{aligned} &\frac{1}{4}+\frac{2}{3} \\\\ &=\frac{3}{12}+\frac{8}{12}\\\\ &=\frac{3+8}{12}\\\\ &=\frac{11}{12} \end{aligned}

### 2. Subtracting fractions

E.g.

\begin{aligned} &\frac{4}{5}-\frac{1}{3}\\\\ &=\frac{12}{15}-\frac{5}{15}\\\\ &=\frac{12-5}{15}\\\\ &=\frac{7}{15} \end{aligned}

### 3. Multiplying fractions

E.g.

\begin{aligned} &\frac{4}{7} \times \frac{2}{5}\\\\ &=\frac{4 \times 2}{7 \times 5}\\\\ &=\frac{8}{35} \end{aligned}

### 4. Dividing fractions

E.g.

\begin{aligned} &\frac{1}{5} \div \frac{3}{4}\\\\ &=\frac{1}{5} \times \frac{4}{3}\\\\ &=\frac{1 \times 4}{5 \times 3}\\\\ &=\frac{4}{15} \end{aligned}

### 5. Equivalent fractions

E.g.

$\frac{10}{40}=\frac{1 \times 10}{4 \times 10}=\frac{1}{4}$

### 6. Improper fractions and mixed numbers

An improper fraction is a fraction where the numerator (top number) is larger than the denominator (bottom number).

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

E.g.

$\frac{21}{5}=\frac{4 \times 5 +1}{5}=4\frac{1}{5}$

### 7. Ordering fractions

E.g.

Write these fractions in order of size from smallest to largest:

\begin{aligned} &\frac{5}{6} \quad \quad \quad \frac{2}{3} \quad \quad \quad \frac{1}{2} \quad \quad \quad \frac{3}{4}\\\\ &\frac{20}{24} \quad \quad \; \frac{16}{24} \quad \quad \; \frac{12}{24} \quad \quad \;\; \frac{18}{24}\\\\ &\frac{12}{24} \quad \quad \; \frac{16}{24} \quad \quad \; \frac{18}{24} \quad \quad \;\; \frac{20}{24} \\\\ &\frac{1}{2} \quad \quad \quad \frac{2}{3} \quad \quad \quad \frac{3}{4} \quad \quad \quad \frac{5}{6} \end{aligned}

### 8. Fractions of an amount

E.g.

Calculate \frac{3}{4} of 28:

\begin{aligned} &\frac{3}{4} \; \text{ of } \; 28 \\\\ &\frac{1}{4} \; \text{ of } \; 28 = 28 \div 4 = 7 \\\\ &\frac{3}{4} \; \text{ of } \; 28 = 3 \times 7 = 21 \end{aligned}

## Decimals

E.g.

$1.37+ 2.8$

Use the column method.

### 2. Subtracting decimals

E.g.

$3.72 – 2.9$

Use the column method.

### 3. Multiplying decimals

E.g.

$2.3 \times 1.7$

2.3 \times 1.7 becomes 23 \times 17

391 becomes 3.91

### 4. Dividing decimals

E.g.

\begin{aligned} &6.3 \div 0.3\\\\ &6.3 \div 0.3 = \frac{6.3}{0.3}=\frac{6.3 \times 10}{0.3 \times 10}=\frac{63}{3}=21 \end{aligned}

## Percentages

### 1. Percentage of an amount

E.g.
Find 35\% of 400

\begin{aligned} 100\% &= 400 \\\\ 10\%&=40\\\\ 5\%&=20 \end{aligned}

So,

$35\% = (3 \times 40) + 20=140$

### 2. Percentage multipliers

E.g.
Find 27\% of 320

$320 \times 0.27 = 86.4$

The multiplier for 27\% is 0.27

### 3. Percentage increase

E.g.
Increase 40 by 12\%

Either find 12\% of 40 and add it on to 40 , or use a multiplier.

\begin{aligned} &100\% + 12\% = 112\% \\\\ &40 \times 1.12 = 44.8 \end{aligned}

### 4. Percentage decrease

E.g.
Decrease 90 by 23\% .

Either find 23\% of 90 and subtract it from 90 , or use a multiplier.

\begin{aligned} &100\% – 23\% = 77\% \\\\ &90 \times 0.77=69.3 \end{aligned}

### 5. Percentage change

E.g.
Calculate the percentage change from 200 to 240 .

\begin{aligned} &\text{Percentage change} = \frac{Change}{Original} \times 100 \\\\ &\frac{240-200}{200} \times 100 = \frac{40}{200} \times 100 = 20 \end{aligned}

Therefore the percentage change is 20\% .

### 6. Reverse percentages

E.g.
80\% of a number is 240 . What was the original number?

## Comparing fractions, decimals and percentages

### 1. Fractions to decimals

Write \frac{5}{8} as a decimal.

Divide the numerator by the decimal by using a written method or a calculator.

$\frac{5}{8}=0.625$

### 2. Decimals to fractions

Write 0.34 as a fraction.

$\frac{0.34}{1}=\frac{0.34\times 100}{1\times 100}=\frac{34}{100}$

Then cancel so that the fraction is in its simplest form.

$\frac{34}{100}=\frac{2 \times 17}{2 \times 50}=\frac{17}{50}$

### 3. Fractions to percentages

Write \frac{7}{8} as a percentage.

$\frac{7}{8}\times 100=\frac{7\times 100}{8}=\frac{700}{8}$

$\frac{7}{8}=87.5\%$

### 4. Percentages to fractions

Write 56\% as a fraction.

\begin{aligned} &\frac{56}{100}=\frac{4\times 14}{4\times 25}=\frac{14}{25} \\\\ &56\% = \frac{14}{25} \end{aligned}

### 5. Decimal to percentage

Write 0.63 as a percentage.

\begin{aligned} &0.63\times 100 = 63 \\\\ &0.63=63\% \end{aligned}

### 6. Percentage to decimal

Write 32\% as a decimal.

\begin{aligned} &32\div 100 = 0.32 \\\\ &32\%=0.32 \end{aligned}

### Common misconceptions

• Common denominators

To be able to add, subtract or compare fractions they must have a common denominator. To do this you need to find a common multiple for the denominators. The lowest common denominator is the easiest to use.

• Fractions in their simplest form

Often fraction questions ask for the answer to be in its simplest form. This means you need to consider the numerator (the top number) and the denominator (the bottom number) and cancel by looking for common factors.

• Percentages can be greater than 100

Percentages can be more than 100 . This can happen for a percentage increase and for calculating percentage change.

• The equivalence of one-third

Take care with one-third and its decimal and percentage equivalence.

\begin{aligned} &\frac{1}{3}=0.3333333… = 33.33333…\%=33\frac{1}{3}\% \\\\ &33\% = 0.33 = \frac{33}{100} \end{aligned}

### Practice fractions, decimals and percentage questions

1.  Calculate:

\frac{7}{8} \; − \; \frac{2}{5}

\frac{19}{40} \frac{5}{3} 1\frac{11}{40} \frac{23}{40} \begin{aligned} &\frac{7}{8}-\frac{2}{5} \\\\ &=\frac{35}{40}-\frac{16}{40}\\\\ &=\frac{35-16}{40}\\\\ &=\frac{19}{40} \end{aligned}

2. Convert the following mixed number to an improper fraction:

2\frac{3}{5}

\frac{23}{5} \frac{6}{10} \frac{13}{5} \frac{11}{5} \begin{aligned}2 \times 5 = 10\\\\ 10 + 3 = 13 \end{aligned}

3. Calculate:

2.8 \times 1.3

3.84 3.64 36.4 38.4  4. Calculate:

2.24 \div 0.4

0.56 56 560 5.6  5.  Increase 45 by 12\%

50.8 50.4 39.6 38.2 \begin{aligned} &100\% + 12\% = 112\% \\\\ &45\times 1.12=50.4 \end{aligned}

6. 65\% of a number is 520 . What is the original number?

800 858 338 700  ### Fractions, decimals and percentages GCSE questions

1.  (a) Write \frac{3}{4} as a decimal

(b) Write 0.7 as a fraction

(c) Write 0.6 as a percentage

(3 marks)

(a) \frac{3}{4}=0.75

(1)

(b) 0.7=\frac{7}{10}

(1)

(c) 0.6=\frac{6}{10}=\frac{60}{100}=60\%

(1)

The cost of the car is £13 600 plus VAT at 20\%

Gordon pays a deposit of £4000

He pays the rest in 10 equal payments.

Work out the amount of each of the 10 payments.

(4 marks)

20\% = 2720

(for finding 20\% of the price)

(1)

120\% = 16320

(for finding 120\% of the price)

(1)

16320-4000=12320

(for finding calculating the remainder to be paid)

(1)

12320\div 10=1232

(for finding calculating the remainder to be paid)

(1)

3. Prove algebraically that the recurring decimal 0.4\dot{3}\dot{2} has the value of \frac{214}{495}

(3 marks)

x=0.4323232…

(for the correct recurring decimal)

(1)

\begin{aligned} &100x=43.232323…. \\\\ &99x=42.8 \end{aligned}

(for the second recurring decimal and the subtraction)

(1)

x=\frac{42.8}{99}=\frac{428}{990}=\frac{214}{495}

(for the correct fraction)

(1)

## Learning checklist

You have now learned how to:

• Order decimals and fractions
• Understand and use place value for decimals
• Use the 4 operations, including formal written methods, applied to decimals, proper and improper fractions, and mixed numbers
• Define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare 2 quantities using percentages, and work with percentages greater than 100\%
• Convert fractions, decimals and percentages

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