Fractions of an Amount

What you'll learn

Find unit fractions, such as $16\frac{1}{6}$ of an amount.
Find non-unit fractions, such as $34\frac{3}{4}$ of an amount.
Work backwards when you know a fraction of a number.
Tackle word problems involving differences, discounts and “the rest”.

The basics: what the fraction is asking

When a question says “find a fraction of an amount”, the amount is the number, price or group you start with.

For example, in $35\frac{3}{5}$ of 100, the fraction is $35\frac{3}{5}$ and the amount is 100.

$The whole amount 100 is split into 5 equal parts, with 3 parts selected for \frac{3}{5}.$

Definition

Parts of a fraction

The denominator is the bottom number. It tells you how many equal parts the amount is split into.
The numerator is the top number. It tells you how many of those equal parts you want.
The amount is the whole number, price or group you start with.

Key Idea

The main method

To find a fraction of an amount, divide by the denominator first, then multiply by the numerator.

Unit fractions: divide by the bottom number

A unit fraction is a fraction with numerator 1, such as $14\frac{1}{4}$ or $18\frac{1}{8}$ . You only need to divide.

Example

Find a unit fraction

Find $16\frac{1}{6}$ of 360.

$360 is divided into 6 equal parts, and one part is taken for \frac{1}{6}.$

The denominator is 6, so split 360 into 6 equal parts.
Divide 360 by 6.
$360 \div 6 = 60$
The numerator is 1, so you take one part.
$16\frac{1}{6}$ of 360 is 60.

Tip

Quick size check

For these questions, the answer should be smaller than the starting amount. If you find $16\frac{1}{6}$ of 360 and get more than 360, check your division.

Non-unit fractions: divide, then multiply

A non-unit fraction has a numerator bigger than 1, such as $34\frac{3}{4}$ or $56\frac{5}{6}$ .

You still divide by the denominator first. Then you multiply by the numerator.

Example

Find a non-unit fraction

Work out $58\frac{5}{8}$ of 96.

96 is split into 8 equal parts, then 5 of those parts are selected.

The denominator is 8, so first find one eighth of 96.
$96 \div 8 = 12$
The numerator is 5, so multiply one eighth by 5.
$12 \times 5 = 60$
So $58\frac{5}{8}$ of 96 is 60.

Common Mistake

Multiplying by the bottom number

For $34\frac{3}{4}$ of an amount, do not multiply by 4. The denominator tells you how many equal parts to split into, so you divide by 4 first.

Working backwards from a fraction

Sometimes you are told the result of a fraction of a hidden number.

A variable is a letter, such as $n$ , that stands for an unknown number.

If $25\frac{2}{5}$ of a number is 26, then 26 represents 2 equal parts. You can work backwards to find all 5 parts.

Example

Find the unknown number

$25\frac{2}{5}$ of a number $n$ is 26. Find $n$ .

The known value 26 represents 2 of the 5 equal parts of the unknown whole n.

The result 26 represents 2 equal parts.
Find one equal part by dividing by 2.
$26 \div 2 = 13$
The whole number has 5 equal parts, so multiply by 5.
$13 \times 5 = 65$
So $n = 65$ .

Difference questions

The difference between two values means the gap between them. Find each value first, then subtract the smaller from the larger.

Percent means “out of 100”. A useful one to remember is that 20% is the same as $15\frac{1}{5}$ .

Example

Find a difference involving a percentage

Work out the difference between 20% of 85 and $34\frac{3}{4}$ of 40.

Two separate amounts are found first, then the gap between them is compared.

Change 20% into a fraction you can use: 20% is $15\frac{1}{5}$ . Find $15\frac{1}{5}$ of 85.
$85 \div 5 = 17$
Find $34\frac{3}{4}$ of 40.
$40÷4=1010×3=30\begin{aligned} 40 \div 4 &= 10\\ 10 \times 3 &= 30 \end{aligned}$
Subtract the smaller value from the larger value.
$30 - 17 = 13$
The difference is 13.

Common Mistake

Stopping too early

In a difference question, finding the fraction is not the final answer. You must do the subtraction as well.

Word problems with money, “off” and “the rest”

In word problems, underline the important phrases:

Reduced by or off means a discount is taken away from the original price.
The rest means what is left after subtracting the parts already used.
Of the children, of the boys, or of the girls means find that group first, then take the fraction.

Sale prices

A discount is an amount taken off a price.

Example

Price reduced in a sale

The normal price of a game is £35. It is reduced by $15\frac{1}{5}$ in a sale. Work out the sale price.

The discount is one fifth of the original £35, and the sale price is what remains.

Find the fraction taken off. Ignore the £ sign for the calculation.
$35 \div 5 = 7$
The discount is £7.
Subtract the discount from the normal price: £35 - £7 = £28.
The sale price is £28.

Groups inside groups

Be careful when a fraction is only about one group, not the whole total.

Example

Find the group first

At a school event there are 300 people. There are 124 adults and 92 teenagers. The rest are children. $37\frac{3}{7}$ of the children are boys. Work out how many girls there are.

Find the children from the total first, then split the children into boys and girls.

First find the number of children.
$300 - 124 - 92 = 84$
Find $37\frac{3}{7}$ of the children to get the number of boys.
$84÷7=1212×3=36\begin{aligned} 84 \div 7 &= 12\\ 12 \times 3 &= 36 \end{aligned}$
The girls are the children who are not boys.
$84 - 36 = 48$
There are 48 girls.

Exam technique

In the exam

Look at the denominator first: that tells you what to divide by.
If the numerator is bigger than 1, multiply after dividing.
In word problems, decide whether you need to subtract at the end, especially when you see “difference”, “off”, “reduced by” or “the rest”.

Self review

Check yourself

What do you divide by when finding $47\frac{4}{7}$ of an amount?
If $35\frac{3}{5}$ of a number is 21, what would you do first?
In a sale question, do you give the discount as the answer, or subtract it from the original price?

Fractions of an Amount

What you'll learn

Find unit fractions, such as $16\frac{1}{6}$ of an amount.
Find non-unit fractions, such as $34\frac{3}{4}$ of an amount.
Work backwards when you know a fraction of a number.
Tackle word problems involving differences, discounts and “the rest”.

The basics: what the fraction is asking

When a question says “find a fraction of an amount”, the amount is the number, price or group you start with.

For example, in $35\frac{3}{5}$ of 100, the fraction is $35\frac{3}{5}$ and the amount is 100.

$The whole amount 100 is split into 5 equal parts, with 3 parts selected for \frac{3}{5}.$

Definition

Parts of a fraction

The denominator is the bottom number. It tells you how many equal parts the amount is split into.
The numerator is the top number. It tells you how many of those equal parts you want.
The amount is the whole number, price or group you start with.

Key Idea

The main method

To find a fraction of an amount, divide by the denominator first, then multiply by the numerator.

Unit fractions: divide by the bottom number

A unit fraction is a fraction with numerator 1, such as $14\frac{1}{4}$ or $18\frac{1}{8}$ . You only need to divide.

Example

Find a unit fraction

Find $16\frac{1}{6}$ of 360.

$360 is divided into 6 equal parts, and one part is taken for \frac{1}{6}.$

The denominator is 6, so split 360 into 6 equal parts.
Divide 360 by 6.
$360 \div 6 = 60$
The numerator is 1, so you take one part.
$16\frac{1}{6}$ of 360 is 60.

Tip

Quick size check

For these questions, the answer should be smaller than the starting amount. If you find $16\frac{1}{6}$ of 360 and get more than 360, check your division.

Non-unit fractions: divide, then multiply

A non-unit fraction has a numerator bigger than 1, such as $34\frac{3}{4}$ or $56\frac{5}{6}$ .

You still divide by the denominator first. Then you multiply by the numerator.

Example

Find a non-unit fraction

Work out $58\frac{5}{8}$ of 96.

96 is split into 8 equal parts, then 5 of those parts are selected.

The denominator is 8, so first find one eighth of 96.
$96 \div 8 = 12$
The numerator is 5, so multiply one eighth by 5.
$12 \times 5 = 60$
So $58\frac{5}{8}$ of 96 is 60.

Common Mistake

Multiplying by the bottom number

For $34\frac{3}{4}$ of an amount, do not multiply by 4. The denominator tells you how many equal parts to split into, so you divide by 4 first.

Working backwards from a fraction

Sometimes you are told the result of a fraction of a hidden number.

A variable is a letter, such as $n$ , that stands for an unknown number.

If $25\frac{2}{5}$ of a number is 26, then 26 represents 2 equal parts. You can work backwards to find all 5 parts.

Example

Find the unknown number

$25\frac{2}{5}$ of a number $n$ is 26. Find $n$ .

The known value 26 represents 2 of the 5 equal parts of the unknown whole n.

The result 26 represents 2 equal parts.
Find one equal part by dividing by 2.
$26 \div 2 = 13$
The whole number has 5 equal parts, so multiply by 5.
$13 \times 5 = 65$
So $n = 65$ .

Difference questions

The difference between two values means the gap between them. Find each value first, then subtract the smaller from the larger.

Percent means “out of 100”. A useful one to remember is that 20% is the same as $15\frac{1}{5}$ .

Example

Find a difference involving a percentage

Work out the difference between 20% of 85 and $34\frac{3}{4}$ of 40.

Two separate amounts are found first, then the gap between them is compared.

Change 20% into a fraction you can use: 20% is $15\frac{1}{5}$ . Find $15\frac{1}{5}$ of 85.
$85 \div 5 = 17$
Find $34\frac{3}{4}$ of 40.
$40÷4=1010×3=30\begin{aligned} 40 \div 4 &= 10\\ 10 \times 3 &= 30 \end{aligned}$
Subtract the smaller value from the larger value.
$30 - 17 = 13$
The difference is 13.

Common Mistake

Stopping too early

In a difference question, finding the fraction is not the final answer. You must do the subtraction as well.

Word problems with money, “off” and “the rest”

In word problems, underline the important phrases:

Reduced by or off means a discount is taken away from the original price.
The rest means what is left after subtracting the parts already used.
Of the children, of the boys, or of the girls means find that group first, then take the fraction.

Sale prices

A discount is an amount taken off a price.

Example

Price reduced in a sale

The normal price of a game is £35. It is reduced by $15\frac{1}{5}$ in a sale. Work out the sale price.

The discount is one fifth of the original £35, and the sale price is what remains.

Find the fraction taken off. Ignore the £ sign for the calculation.
$35 \div 5 = 7$
The discount is £7.
Subtract the discount from the normal price: £35 - £7 = £28.
The sale price is £28.

Groups inside groups

Be careful when a fraction is only about one group, not the whole total.

Example

Find the group first

At a school event there are 300 people. There are 124 adults and 92 teenagers. The rest are children. $37\frac{3}{7}$ of the children are boys. Work out how many girls there are.

Find the children from the total first, then split the children into boys and girls.

First find the number of children.
$300 - 124 - 92 = 84$
Find $37\frac{3}{7}$ of the children to get the number of boys.
$84÷7=1212×3=36\begin{aligned} 84 \div 7 &= 12\\ 12 \times 3 &= 36 \end{aligned}$
The girls are the children who are not boys.
$84 - 36 = 48$
There are 48 girls.

Exam technique

In the exam

Look at the denominator first: that tells you what to divide by.
If the numerator is bigger than 1, multiply after dividing.
In word problems, decide whether you need to subtract at the end, especially when you see “difference”, “off”, “reduced by” or “the rest”.

Self review

Check yourself

What do you divide by when finding $47\frac{4}{7}$ of an amount?
If $35\frac{3}{5}$ of a number is 21, what would you do first?
In a sale question, do you give the discount as the answer, or subtract it from the original price?

Fractions of an Amount

What you'll learn

The basics: what the fraction is asking

Unit fractions: divide by the bottom number

Non-unit fractions: divide, then multiply

Working backwards from a fraction

Difference questions

Word problems with money, “off” and “the rest”

Sale prices

Groups inside groups

Recap questions

You've reached the end

Fractions of an Amount

What you'll learn

The basics: what the fraction is asking

Unit fractions: divide by the bottom number

Non-unit fractions: divide, then multiply

Working backwards from a fraction

Difference questions

Word problems with money, “off” and “the rest”

Sale prices

Groups inside groups

Recap questions

You've reached the end

Fractions of an Amount

What you'll learn

The basics: what the fraction is asking

Unit fractions: divide by the bottom number

Non-unit fractions: divide, then multiply

Working backwards from a fraction

Difference questions

Word problems with money, “off” and “the rest”

Sale prices

Groups inside groups

Recap questions

You've reached the end

Fractions of an Amount Revision Guide

Fractions of an Amount

What you'll learn

The basics: what the fraction is asking

Unit fractions: divide by the bottom number

Non-unit fractions: divide, then multiply

Working backwards from a fraction

Difference questions

Word problems with money, “off” and “the rest”

Sale prices

Groups inside groups

Recap questions

You've reached the end

Fractions of an Amount Revision Guide