GCSE Maths · Higher · Chapter 2: Fractions, ratio and proportion

Fraction arithmetic and percentage change

Grades 4–9About 80 minutesNo calculator87 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/fractions-ratio-and-proportion/

NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Work out $\frac{2}{3} + \frac{1}{6}$ .
[1 mark]
Answer

A2
Write $\frac{18}{24}$ in its simplest form.
[1 mark]
Answer
A3
Write $2 \frac{3}{5}$ as an improper fraction.
[1 mark]
Answer
A4
Work out 15% of 80.
[1 mark]
Answer
A5
Write $\frac{7}{20}$ as a percentage.
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

Work out $2 \frac{1}{4} + 1 \frac{5}{6}$ .

Hint Write each mixed number as an improper fraction. Then use a common denominator: 12 is the smallest one that 4 and 6 both go into.

B1
Work out $3 \frac{2}{5} - 1 \frac{3}{4}$ .
[3 marks]
Answer

Example 2

Work out $1 \frac{3}{5} \div 2 \frac{2}{3}$ .

Hint Write both as improper fractions. Dividing by a fraction is the same as multiplying by its reciprocal.

B2
Work out $2 \frac{1}{4} \div 1 \frac{2}{7}$ .
[3 marks]
Answer

Example 3

Increase £320 by 15%.

Hint The new amount is 115% of the old one, so multiply by 1.15. Or find 15% and add it on.

B3
Decrease £360 by 35%.
[2 marks]
Answer

Example 4

The price of a ticket rose from £40 to £46. Work out the percentage increase.

Hint Find the change first. Then write the change as a percentage of the ORIGINAL price.

B4
The number of members of a club fell from 250 to 215. Work out the percentage decrease.
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Write 45 minutes as a fraction of 2 hours. Give your answer in its simplest form.
[2 marks]
Answer

C2
Write these numbers in order of size. Start with the smallest.
$\frac{5}{8} 0.61 63% \frac{3}{5}$
[2 marks]
Answer
C3
Work out $\frac{4}{9} \times \frac{15}{16}$ . Give your answer in its simplest form.
[2 marks]
Answer
C4
There are 32 students in a class. $\frac{3}{8}$ of them walk to school, 25% come by bus and the rest come by car.
(a)
How many of the students come by car?
[2 marks]
Answer
(b)
What percentage of the class come by car?
[2 marks]
Answer
Total for C4: 4 marks
C5
Write down the reciprocal of $2 \frac{1}{3}$ .
[1 mark]
Answer
C6
Asha scored 42 out of 56 in her first test and 54 out of 75 in her second test. In which test did she get the higher percentage? You must show your working.
[3 marks]
Answer
C7
Work out $\frac{3}{4} - \frac{2}{3} \times \frac{3}{8}$ .
[2 marks]
Answer
C8
A water tank is $\frac{2}{5}$ full. After 36 litres are added, it is $\frac{7}{10}$ full. How many litres does the tank hold when it is full?
[3 marks]
Answer
C9
Find the number that is exactly halfway between $\frac{2}{3}$ and $\frac{4}{5}$ .
[2 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Show that $1 \frac{5}{6} \div 2 \frac{4}{9} = \frac{3}{4}$ .
[3 marks]

D2
A shop buys a jacket for £80 and sells it for £120.
(a)
Work out the profit as a percentage of the price the shop paid.
[2 marks]
Answer
(b)
In a sale, the shop reduces its selling price of the jacket by 15%. Work out the profit in the sale as a percentage of the price the shop paid.
[3 marks]
Answer
Total for D2: 5 marks
D3
Sam says, " $\frac{2}{5} + \frac{3}{4} = \frac{5}{9}$ ."
Explain why Sam is wrong, and work out the correct answer.
[3 marks]
Answer
D4
Kerry has £2400. She gives $\frac{1}{3}$ of it to her brother. She gives 25% of what is left to her sister. Then she spends $\frac{3}{8}$ of what she has left.
How much money does Kerry have now?
[4 marks]
Answer
D5
Jo's salary rises from £24 000 to £25 080. Show that this is an increase of 4.5%.
[2 marks]
D6
A rectangular garden is $12 \frac{1}{2}$ m long and $8 \frac{2}{5}$ m wide.
(a)
Show that the area of the garden is 105 m².
[2 marks]
(b)
$\frac{3}{7}$ of the garden is to be sown with grass seed. A bag of seed covers 9 m² and costs £4.80. The shop takes 10% off the price when you buy 5 or more bags.
Work out the cost of the seed for the garden.
[4 marks]
Answer
Total for D6: 6 marks
D7
400 people were asked which hot drink they like best. $\frac{3}{8}$ of them said tea. 36% of the rest said coffee. Everyone else said neither.
What percentage of the 400 people said neither?
[4 marks]
Answer
D8
Work out $(2 \frac{1}{3})^{2} - 1 \frac{1}{2} \div \frac{3}{4}$ . Give your answer as a mixed number.
[3 marks]
Answer
D9
Last year a café sold 1250 cups of coffee a week at £2.40 a cup. This year it sells 1400 cups a week at £2.70 a cup.
Work out the percentage increase in the money the café takes for coffee each week.
[4 marks]
Answer
D10
A ribbon is 2.4 m long. Hasan cuts 45 cm off it. Then he cuts off $\frac{2}{5}$ of the ribbon that is left.
What fraction of the original ribbon is left now? Give your answer in its simplest form.
[3 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
A unit fraction has 1 as its numerator, such as $\frac{1}{7}$ .
Find every way of writing $\frac{5}{12}$ as the sum of two different unit fractions.
[4 marks]
Hint 1 · What to try
Call the larger fraction $\frac{1}{a}$ . It must be more than half of $\frac{5}{12}$ , but less than $\frac{5}{12}$ .
Hint 2 · The first line
Half of $\frac{5}{12}$ is $\frac{5}{24}$ , so $\frac{1}{a}$ is between $\frac{5}{24}$ and $\frac{5}{12}$ . That leaves $a = 3$ or $a = 4$ .
Hint 3 · The full method
$\frac{5}{12} - \frac{1}{3} = \frac{1}{12}$ and $\frac{5}{12} - \frac{1}{4} = \frac{2}{12} = \frac{1}{6}$ . So $\frac{1}{3} + \frac{1}{12}$ and $\frac{1}{4} + \frac{1}{6}$ are the only two ways.
Answer

E2
A fraction and its reciprocal add up to $\frac{13}{6}$ . Find the fraction. Give every possible answer.
[3 marks]
Hint 1 · What to try
Write the fraction as $\frac{a}{b}$ , so its reciprocal is $\frac{b}{a}$ . Add them.
Hint 2 · The first line
$\frac{a}{b} + \frac{b}{a} = \frac{a ^{2} + b ^{2}}{ab}$
Hint 3 · The full method
So $a^{2} + b^{2} = 13$ and $ab = 6$ will do: $a = 2$ and $b = 3$ . The fraction is $\frac{2}{3}$ , or its reciprocal $\frac{3}{2}$ , which works just as well.
Answer
E3
In a school, 45% of the students are boys. 20% of the boys and $x$ % of the girls play in the orchestra. Altogether, 25.5% of the students play in the orchestra.
Find the value of $x$ .
[3 marks]
Hint 1 · What to try
Suppose the school has 100 students. Work out how many are boys and how many are girls.
Hint 2 · The first line
$20%$ of $45$ boys is $9$ , and $25.5$ of the $100$ students play, so $16.5$ girls play.
Hint 3 · The full method
$16.5$ out of $55$ girls is $\frac{16.5}{55} \times 100 = 30%$ , so $x = 30$ .
Answer
E4
Work out the exact value of
$(1 - \frac{1}{2}) (1 - \frac{1}{3}) (1 - \frac{1}{4}) \dots (1 - \frac{1}{50})$
[3 marks]
Hint 1 · What to try
Work out each bracket as a single fraction.
Hint 2 · The first line
$\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \dots \times \frac{49}{50}$
Hint 3 · The full method
Each numerator cancels with the denominator before it. Only the first numerator, 1, and the last denominator, 50, are left. The answer is $\frac{1}{50}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can add, subtract, multiply and divide fractions and mixed numbers.
I can write one quantity as a fraction of another.
I can increase or decrease an amount by a percentage.
I can work out a percentage change and write one quantity as a percentage of another.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers