Fractions of Amounts: Edexcel GCSE Foundation maths revision notes

Start with one equal part

In \frac{3}{5}, the bottom number, 5, is the denominator. It tells you to split the whole amount into five equal parts. The top number, 3, is the numerator. It tells you to take three of those parts.

A unit fraction has numerator 1, such as \frac{1}{6}. To find \frac{1}{6} of an amount, divide the amount by 6.

In the exam you are usually asked to work out a fraction of an amount, such as "Work out \frac{2}{3} of 45", or to write one amount as a fraction of another, for 1 or 2 marks. Longer questions, worth 3 to 5 marks, take a fraction of an amount and then a fraction of what is left.

Worked example 1: Finding one sixth

Work out \frac{1}{6} of 42

One sixth means one of six equal parts, so divide by 6
Your turn. What is 42\div6?

Your answer

42\div6=7
Check: six parts of 7 make 6\times7=42, the whole amount.
Answer: 7

Take the right number of parts

For a fraction such as \frac{3}{5}, first find one fifth, then take three of those fifths. You can write the method in words.

\text{amount}\div\text{denominator}\times\text{numerator}

Dividing first keeps the numbers small. On a calculator paper you can also type the fraction times the amount, because "of" means multiply.

Worked example 2: Three fifths of an amount

Work out \frac{3}{5} of 80

The denominator is 5, so split 80 into five equal parts.
Your turn. Work out 80\div5

Your answer

80\div5=16, so one fifth of 80 is 16
The bar shows 80 split into five parts of 16. The numerator is 3, so take three parts.
3\times16=48
Answer: 48

Worked example 3: A fraction of money

Imani gives \frac{5}{8} of £36 to a charity. How much money does she give?

One eighth: 36\div8=4.5, which is £4.50.
Your turn. Five eighths is five of these parts. Work out 4.50\times5

Your answer

4.50\times5=22.50
The answer is an amount of money, so write it in pounds with two decimal places.
Answer: £22.50

Write one amount as a fraction of another

To write one amount as a fraction of a total, put the part on top and the whole on the bottom.

\text{fraction}=\frac{\text{part}}{\text{whole}}

Then simplify: divide the top and the bottom by the same number until no number other than 1 divides both. For example, \frac{12}{20}=\frac{3}{5}, dividing both by 4.

Both amounts must be in the same units. 30 cm out of 2 metres is 30 cm out of 200 cm, so the fraction is \frac{30}{200}=\frac{3}{20}, not \frac{30}{2}.

Worked example 4: What fraction is left?

A cinema has 120 seats. 75 of the seats are taken. What fraction of the seats are empty? Give your fraction in its simplest form.

The question asks about the empty seats, so first find how many seats are empty.
Your turn. Work out 120-75

Your answer

Empty seats: 120-75=45
The whole is all 120 seats, so the fraction is \frac{45}{120}
Divide the top and the bottom by 15: \frac{45}{120}=\frac{3}{8}
Answer: \frac{3}{8}

Work backwards to the whole

Sometimes you are told a fraction of an amount and must find the whole amount. If \frac{1}{7} of a number is 8, the whole is seven parts of 8: 8\times7=56

When more than one part is given, divide by the numerator to find one part, then multiply by the denominator to find the whole. This undoes the forward method.

Read carefully which fraction you have been given. If Ola spends \frac{2}{5} of her money and has £30 left, the £30 is the \frac{3}{5} that is left, so one fifth is £10 and she started with £50.

Worked example 5: Finding the whole amount

\frac{3}{4} of a number is 27. Work out the number.

Three quarters is three equal parts, and together they make 27.
Your turn. If three equal parts make 27, what is one part?

Your answer

One part is 27\div3=9
The whole is four parts: 9\times4=36
Check forwards: \frac{3}{4} of 36 is 36\div4\times3=27
Answer: 36

Be clear which amount is the whole

If \frac{2}{5} of an amount is used, the fraction left is 1-\frac{2}{5}=\frac{3}{5}. You can work out the part used and subtract it, or find \frac{3}{5} of the amount straight away.

When a question says "of what is left" or "of the rest", the whole has changed. Work out what is left first, then take the fraction of that new amount.

Worked example 6: A fraction of what is left

A farmer has 450 kg of potatoes. On Monday he sells \frac{1}{3} of the potatoes. On Tuesday he sells \frac{3}{5} of the potatoes he has left. How many kilograms of potatoes does he have left now?

Sold on Monday: 450\div3=150 kg
Left after Monday: 450-150=300 kg
Your turn. Tuesday's fraction is of the 300 kg left, not of 450 kg. Work out \frac{3}{5} of 300

Your answer

Sold on Tuesday: 300\div5\times3=180 kg
Left after Tuesday: 300-180=120 kg
Answer: 120 kg

Compare and decide

Some questions ask you to decide, for example "Who gets more?" or "Is Sam correct?". Work out each actual amount, then write a short sentence that uses your numbers.

A bigger fraction does not always give a bigger amount when the totals are different. \frac{1}{2} of 60 is 30, but \frac{2}{3} of 36 is only 24.

Common mistakes

Dividing by the numerator. To find \frac{3}{5} of an amount, divide by 5, the denominator, then multiply by 3.
Taking a fraction of the wrong amount. When a fraction is "of what is left", work out what is left first and use that.
Stopping one step too soon. If the question asks how much is left, subtract the part used from the whole.
Putting the wrong number on the bottom. The denominator is the whole group, and the top is the part the question asks about.
Not simplifying fully. Dividing by 2 once is often not enough: keep dividing until no number other than 1 divides the top and the bottom.
Giving a decimal or a percentage. If the question asks "What fraction?", give a fraction.