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Place value Arithmetic

This topic is relevant for:

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?

$What are fractions, decimals and percentages?$

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: $\frac{4}{9}$

Fractions, Decimals and Percentages image 1

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 $\frac{7}{10}$

Fractions, Decimals and Percentages image 2

$\frac{7}{10}$ 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.

Fractions, Decimals and Percentages image 3

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, decimals and percentages worksheet

Get your free fractions, decimals and percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Fractions, decimals and percentages worksheet

Get your free fractions, decimals and percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Fractions, decimals and percentages examples

Fractions

1. Adding fractions

To add fractions they need to have a common denominator.

E.g.

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

Step-by-step guide: Adding fractions

2. Subtracting fractions

To subtract fractions they need to have a common denominator.

E.g.

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

Step-by-step guide: Subtracting fractions

3. Multiplying fractions

To multiply fractions we need to multiply the numerators together and multiply the denominators together.

E.g.

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

Step-by-step guide: Multiplying fractions

4. Dividing fractions

To divide fractions we need to find the reciprocal of (flip) the second fraction, change the divide sign to a multiply and then multiply the fractions together.

E.g.

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

Step-by-step guide: Dividing fractions

5. Equivalent fractions

Equivalent fractions are fractions that have the same value.

E.g.

\[ \frac{10}{40}=\frac{1 \times 10}{4 \times 10}=\frac{1}{4}\]

Step-by-step guide: Equivalent fractions

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.

We can convert between improper fractions and mixed numbers:

E.g.

\[ \frac{21}{5}=\frac{4 \times 5 +1}{5}=4\frac{1}{5}\]

Step-by-step guide: Improper fractions and mixed numbers

7. Ordering fractions

To order fractions they need to have a common denominator.

E.g.

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

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

Step-by-step guide: Ordering fractions

8. Fractions of an amount

We can calculate a fraction of a given amount.

E.g.

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

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

Step-by-step guide: Fractions amounts

Decimals

1. Adding decimals

We can add decimals together:

E.g.

\[1.37+ 2.8\]

Fractions, Decimals and Percentages image 4

Use the column method.

Step-by-step guide: Adding decimals

2. Subtracting decimals

We can subtract decimals from each other:

E.g.

\[3.72 – 2.9\]

Fractions, Decimals and Percentages image 5

Use the column method.

Step-by-step guide: Subtracting decimals

3. Multiplying decimals

We can multiply decimals:

E.g.

\[2.3 \times 1.7\]

$2.3 \times 1.7$ becomes $23 \times 17$

Fractions, Decimals and Percentages image 7

$391$ becomes $3.91$

Step-by-step guide: Multiplying decimals

4. Dividing decimals

We can divide decimals by using equivalent fractions to ensure that the divisor (the denominator) is an integer:

E.g.

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

Step-by-step guide: Dividing decimals

Percentages

1. Percentage of an amount

We can find a percentage of an amount by breaking the percentage down:

E.g.
Find $35\%$ of $400$

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

So,

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

Step-by-step guide: Percentage of an amount

2. Percentage multipliers

We can use percentage multipliers to find a percentage of an amount or to increase/decrease by a percentage:

E.g.
Find $27\%$ of $320$

\[320 \times 0.27 = 86.4\]

The multiplier for $27\%$ is $0.27$

Step-by-step guide: Percentage multipliers

3. Percentage increase

We can increase a value by a percentage:

E.g.
Increase $40$ by $12\%$

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

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

Step-by-step guide: Percentage increase

4. Percentage decrease

We can decrease a value by a percentage:

E.g.
Decrease $90$ by $23\%$ .

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

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

Step-by-step guide: Percentage decrease

5. Percentage change

We can calculate the percentage change between two values:

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 $20\%$ .

Step-by-step guide: Percentage change

6. Reverse percentages

We can use reverse percentages to calculate the original number:

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

Fractions, Decimals and Percentages image 8

Step-by-step guide: Reverse percentages

Comparing fractions, decimals and percentages

1. Fractions to decimals

Converting fractions to decimals:

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

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

Fractions, Decimals and Percentages image 9

\[\frac{5}{8}=0.625\]

Step-by-step guide: Fractions to decimals

2. Decimals to fractions

Converting 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.

\[\frac{34}{100}=\frac{2 \times 17}{2 \times 50}=\frac{17}{50}\]

Step-by-step guide: Decimals to fractions

3. Fractions to percentages

Converting fractions to percentages:

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

\[\frac{7}{8}\times 100=\frac{7\times 100}{8}=\frac{700}{8}\]

Fractions, Decimals and Percentages image 10

\[\frac{7}{8}=87.5\%\]

Step-by-step guide: Fractions to percentages

4. Percentages to fractions

Converting percentages to fractions:

Write $56\%$ as a fraction.

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

Step-by-step guide: Percentages to fractions

5. Decimal to percentage

Converting decimals to percentages:

Write $0.63$ as a percentage.

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

Step-by-step guide: Decimal to percentage

6. Percentage to decimal

Converting percentages to decimals:

Write $32\%$ as a decimal.

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

Step-by-step guide: Percentage to decimal

7. Recurring decimals to fractions

Converting recurring decimals to fractions:

Fractions, Decimals and Percentages image 11

Step-by-step guide: Recurring decimals to fractions

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

\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}

\frac{23}{5}

\frac{6}{10}

\frac{13}{5}

\frac{11}{5}

\begin{aligned}2 \times 5 = 10\\\\ 10 + 3 = 13 \end{aligned}

3.84

3.64

36.4

38.4

Fractions, Decimals and Percentages practice question 3

0.56

56

560

5.6

Fractions, Decimals and Percentages practice question 4

50.8

50.4

39.6

38.2

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

800

858

338

700

Fractions, Decimals and Percentages practice question 6

Fractions, decimals and percentages GCSE questions

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

(b) Write $0.7$ as a fraction

(3 marks)

Show answer

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

(1)

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

(1)

2. Gordon buys a car.

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)

Show answer

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)

Show answer

x=0.4323232\dots

(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

The next lessons are

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