GCSE Maths · Higher · Chapter 5: Ratio and proportion

Ratio, rates and repeated percentages

Grades 4–9About 85 minutesCalculator allowed83 marks

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NameClassDate

Before you start

Grades 4–5

You will need these in every question below.

A1
Write the ratio $18 : 45$ in its simplest form.
[1 mark]
Answer

A2
Write down the multiplier for an increase of 7%.
[1 mark]
Answer
A3
Write 2 hours 15 minutes in hours, as a decimal.
[1 mark]
Answer
A4
Share £60 in the ratio $1 : 4$ .
[2 marks]
Answer
A5
Work out 12% of 350.
[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

Share £720 in the ratio $3 : 5 : 7$ .

Hint Add the parts to find how many equal shares there are. Work out one share, then multiply.

B1
Share 840 g in the ratio $2 : 3 : 9$ .
[2 marks]
Answer

Example 2

A 400 g jar of coffee costs £2.36. A 650 g jar costs £3.77. Which jar is better value?

Hint Work out the cost of 1 g for each jar. The jar with the lower cost for each gram is better value.

B2
A 250 ml bottle of shampoo costs £1.45. A 750 ml bottle costs £4.20. Which bottle is better value? You must show your working.
[3 marks]
Answer

Example 3

A cyclist rides 42 km in 1 hour 45 minutes. Work out her average speed in km/h.

Hint Write the time in hours first: 45 minutes is

\frac{45}{60} = 0.75

of an hour. Then speed

=

distance

\div

time.

B3
A runner covers 21 km in 1 hour 24 minutes. Work out his average speed in km/h.
[2 marks]
Answer

Example 4

£4500 is invested for 4 years at 3.5% compound interest per year. Work out the value of the investment after 4 years.

Hint The multiplier for an increase of 3.5% is 1.035. Multiplying by it once for each year is the same as multiplying by

1.03 5^{4}

B4
A car is worth £18 000. Its value falls by 12% each year. Work out its value after 3 years.
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
6 pens cost £4.86. Work out the cost of 11 of the same pens.
[2 marks]
Answer

C2
The density of gold is 19.3 g/cm³.
(a)
Work out the mass of a gold bar with a volume of 12 cm³.
[2 marks]
Answer
(b)
Work out the volume of 500 g of gold. Give your answer to 3 significant figures.
[2 marks]
Answer
Total for C2: 4 marks
C3
A box has a weight of 360 N. It rests on a face that measures 0.8 m by 0.6 m. Work out the pressure on the floor, in N/m².
[2 marks]
Answer
C4
In a sale, all prices are reduced by 15%. A jacket costs £71.40 in the sale. Work out the price of the jacket before the sale.
[3 marks]
Answer
C5
Amy and Ben share some money in the ratio $3 : 10$ . Ben gets £63 more than Amy. How much money do they share altogether?
[3 marks]
Answer
C6
A train travels at 72 km/h. Write this speed in metres per second.
[2 marks]
Answer
C7
The number of birds in a colony is 8000. The number falls by 6% each year. After how many whole years will the number first be below 6000?
[3 marks]
Answer
C8
A recipe for 6 people uses 450 g of flour. How much flour is needed for 14 people?
[2 marks]
Answer
C9
A bag holds 65 counters. The ratio of red counters to blue counters is $4 : 9$ .
(a)
What fraction of the counters are red?
[1 mark]
Answer
(b)
How many of the counters are blue?
[1 mark]
Answer
Total for C9: 2 marks

Exam-style questions

Grades 6–9

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

D1
Mo says, "If a price goes up by 20% and then goes down by 20%, it ends up back where it started."
Explain why Mo is wrong.
[2 marks]

D2
Lena drives 120 km at an average speed of 80 km/h. She then drives 90 km at an average speed of 60 km/h.
Work out her average speed for the whole journey.
[4 marks]
Answer
D3
An alloy is made by melting together 300 cm³ of metal A and 200 cm³ of metal B. Metal A has density 8.9 g/cm³. Metal B has density 2.7 g/cm³.
Work out the density of the alloy.
[4 marks]
Answer
D4
Priya has £6000 to invest for 4 years.
Account A pays 3.8% compound interest each year.
Account B pays 5% interest in the first year, then 3% compound interest in each year after that.
Which account gives Priya more money after 4 years, and by how much? You must show your working.
[6 marks]
Answer
D5
Some money is invested at 4% compound interest per year. After 3 years the investment is worth £4049.51. How much money was invested?
[3 marks]
Answer
D6
A force acts on an area of 0.5 m² and makes a pressure of 240 N/m². The same force acts on an area of 0.3 m². Work out the new pressure.
[3 marks]
Answer
D7
In a club, the ratio of boys to girls is $5 : 3$ . Then 12 more girls join the club, and no one leaves. The ratio of boys to girls is now $5 : 4$ .
How many boys are in the club?
[4 marks]
Answer
D8
Shop A sells a pack of 6 cans of drink for £3.54. Shop B sells a pack of 15 of the same cans for £9.60, and takes 10% off the price.
Which shop gives the better value? You must show your working.
[3 marks]
Answer
D9
A train leaves at 09:47. It travels 228 km at an average speed of 95 km/h. At what time does it arrive?
[3 marks]
Answer
D10
Water flows into an empty tank at 12 litres per minute. The tank is a cuboid 80 cm long, 60 cm wide and 50 cm high. How many minutes does it take to fill the tank? (1 litre $= 1000$ cm³.)
[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
Ann, Bob and Cal share £1400. The ratio of Ann's share to Bob's share is $2 : 3$ . The ratio of Bob's share to Cal's share is $4 : 5$ .
How much does each person get?
[3 marks]
Hint 1 · What to try
Bob is in both ratios. Make his number of parts the same in both.
Hint 2 · The first line
$2 : 3 = 8 : 12$ and $4 : 5 = 12 : 15$ , so Ann : Bob : Cal $= 8 : 12 : 15$ .
Hint 3 · The full method
That is 35 parts, so one part is $1400 \div 35 = 40$ . Ann gets £320, Bob £480 and Cal £600.
Answer

E2
The value of a house goes up by the same percentage each year. After 2 years its value has gone up by 21% altogether.
What is the percentage increase each year?
[3 marks]
Hint 1 · What to try
Call the yearly multiplier $m$ . Two years of growth multiply the value by $m^{2}$ .
Hint 2 · The first line
A rise of 21% altogether means $m^{2} = 1.21$ .
Hint 3 · The full method
$m = 1.21 = 1.1$ , so the value goes up by 10% each year.
Answer
E3
If Jo walks to school at 5 km/h, she arrives 6 minutes late. If she walks at 6 km/h, she arrives 2 minutes early.
How far is it from Jo's home to school?
[3 marks]
Hint 1 · What to try
Call the distance $d$ km. Write each journey time in hours, in terms of $d$ .
Hint 2 · The first line
The two times differ by 8 minutes, which is $\frac{8}{60}$ of an hour: $\frac{d}{5} - \frac{d}{6} = \frac{8}{60}$ .
Hint 3 · The full method
$\frac{d}{30} = \frac{8}{60}$ , so $d = 4$ . The distance is 4 km.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can share an amount in a ratio and use the unitary method.
I can compare best buys by working out the cost of one unit.
I can use speed, density and pressure, and change between units of speed.
I can work out repeated percentage change with a multiplier and power.
I can find the original amount after a percentage change.

Answers and mark scheme

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