GCSE Tutoring Programme

"Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring."

This topic is relevant for:

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 youβre still stuck.

**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\%

**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}

**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

**Percentages** are numbers which are expressed as parts of 100 .

Percent means βnumber of parts per hundredβ 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.

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:

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

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

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

DOWNLOAD FREETo **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**

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**

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**

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**

**Equivalent fractions **are fractions that have the **same value**.Β

E.g.

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

**Step-by-step guide: Equivalent fractions**

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**

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**

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**

We can add decimals together:

E.g.

\[1.37+ 2.8\]

Use the column method.

**Step-by-step guide: Adding decimals**

We can subtract decimals from each other:

E.g.

\[3.72 – 2.9\]

Use the column method.

**Step-by-step guide: Subtracting decimals**

We can multiply decimals:

E.g.

\[2.3 \times 1.7\]

2.3 \times 1.7 becomes 23 \times 17

391 becomes 3.91

**Step-by-step guide: Multiplying 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**

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**

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**

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**

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**

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**

We can use reverse percentages to calculate the original number:

E.g.

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

**Step-by-step guide: Reverse percentages**

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.

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

**Step-by-step guide:**Fractions to decimals

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**

Converting fractions to percentages:

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

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

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

**Step-by-step guide: Fractions to percentages**

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**

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**

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**

Converting recurring decimals to fractions:

**Step-by-step guide: Recurring decimals to fractions**

**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.

\[ \begin{aligned}
&\frac{1}{3}=0.3333333β¦ = 33.33333β¦\%=33\frac{1}{3}\% \\\\
&33\% = 0.33 = \frac{33}{100}
\end{aligned}
\]

1.Β Calculate:

\frac{7}{8} \; β \; \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.Β 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

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

(b) Write 0.7 as a fraction

(c) Write 0.6 as a percentage

**(3 marks)**

Show answer

(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)**

2. Gordon buys a car.

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

Gordon pays a deposit of Β£4000

He pays the rest in 10 equal payments.

Work out the amount of each of the 10 payments.

**(4 marks)**

Show answer

20\% =Β 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…

(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)**

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 βnumber of parts per hundredβ, 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

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 tuition programme.