KS3 Maths · Key Stage 3 · Unit 5: Ratio, Percentages and Circles

Ratio, percentage change, circles, trapezia and inequalities

Year 8About 85 minutesCalculator allowed98 marks

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NameClassDate

Before you start

Recall

You will need these in every question below.

A1
Write $\frac{35}{56}$ in its simplest form.
[1 mark]
Answer

A2
Work out $15%$ of £260.
[2 marks]
Answer
A3
A triangle has a base of 9 cm and a perpendicular height of 7 cm. Work out its area.
[2 marks]
Answer
A4
Solve $4 x + 7 = 31$ .
[2 marks]
Answer $x = \dots\dots$
A5
Round $47.256$ to 1 decimal place.
[1 mark]
Answer

Examples, then your turn

Core

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

Example 1

Share £120 in the ratio $3 : 5$ .

Hint Add the numbers in the ratio to find how many equal parts there are. Find one part, then multiply.

B1
Share £162 in the ratio $4 : 5$ .
[3 marks]
Answer£ $\dots\dots$ and £ $\dots\dots$

Example 2

A bike costs £180. Its price goes up by $15%$ . Use a multiplier to work out the new price.

Hint An increase of

15%

leaves

115%

of the price, so the multiplier is

1.15

B2
A phone costs £340. In a sale its price goes down by $25%$ . Use a multiplier to work out the sale price.
[2 marks]
Answer

Example 3

A circular plate has a diameter of 22 cm. Work out its area. Give your answer to 1 decimal place.

Hint The area is

π r^{2}

, and

r

is the radius. Halve the diameter first.

B3
A circular rug has a diameter of 34 cm. Work out its area. Give your answer to 1 decimal place.
[2 marks]
Answer

Example 4

Solve $7 x - 9 = 3 x + 11$ .

Hint Take

3 x

from both sides so the unknown is on one side only. Then add 9 to both sides.

B4
Solve $5 (x - 2) = 2 x + 14$ .
[3 marks]
Answer $x = \dots\dots$

Practice

Core

Each question asks for something different. Show your working.

C1
A 750 g jar of honey costs £4.20. A 450 g jar of the same honey costs £2.61. Which jar is better value? You must show your working.
[3 marks]
AnswerThe $\dots\dots$ g jar

C2
Ali and Ben share some sweets in the ratio $4 : 7$ . Ben gets 18 more sweets than Ali. How many sweets are there altogether?
[3 marks]
Answer
C3
After a $15%$ increase, a train ticket costs £27.60. Work out the price of the ticket before the increase.
[3 marks]
Answer
C4
A car is worth £15 000. Its value falls by $12%$ every year. Work out its value after 3 years.
[3 marks]
Answer
C5
The diagram shows a trapezium.
Not drawn accurately
(a)
Work out the area of the trapezium.
[2 marks]
Answer
(b)
A second trapezium has the same area and a height of 8 cm. One of its parallel sides is 7 cm. Work out the length of the other parallel side.
[2 marks]
Answer
Total for C5: 4 marks
C6
The circumference of a circular pond is 44 m. Work out the diameter of the pond. Give your answer to 1 decimal place.
[2 marks]
Answer
C7
Here is an inequality: $9 - 2 x > 3 x - 11$ .
(a)
Solve the inequality.
[2 marks]
Answer
(b)
Write down every positive whole number that satisfies the inequality.
[1 mark]
Answer
Total for C7: 3 marks
C8
Asha thinks of a number. She multiplies it by 6 and then subtracts 13. She gets the same answer when she multiplies the number by 2 and then adds 15. Work out the number.
[3 marks]
Answer

Test-style questions

Stretch

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

D1
Zoe says, "If a price goes up by $25%$ and then down by $25%$ , it ends up where it started."
A games console costs £320. In March its price goes up by $25%$ . In April the new price goes down by $25%$ . Work out the price in April, and explain why Zoe is wrong.
[3 marks]
Answer£ $\dots\dots$

D2
A fruit drink is made from orange, mango and apple juice in the ratio $5 : 2 : 3$ .
(a)
Lena makes 1.5 litres of the drink. How many millilitres of mango juice does she use?
[2 marks]
Answer
(b)
Kai has only 400 ml of orange juice, and plenty of mango and apple juice. What is the greatest amount of the drink he can make?
[2 marks]
Answer
Total for D2: 4 marks
D3
A running track is made of two straight sides, each 80 m long, and a semicircle at each end. The semicircles have a diameter of 50 m.
Not drawn accurately
(a)
Work out the distance once round the track. Give your answer to 1 decimal place.
[3 marks]
Answer
(b)
The grass inside the track is the rectangle and the two semicircles. Work out the area of the grass. Give your answer to 1 decimal place.
[3 marks]
Answer
Total for D3: 6 marks
D4
A trapezium has parallel sides of length $(x + 3)$ cm and $(2 x - 1)$ cm. The perpendicular height is 6 cm. The area is 60 cm².
(a)
Work out the value of $x$ . You must show your working.
[3 marks]
Answer $x = \dots\dots$
(b)
Write down the lengths of the two parallel sides.
[1 mark]
Answer $\dots\dots$ cm and $\dots\dots$ cm
Total for D4: 4 marks
D5
The rectangle and the square have the same perimeter. All lengths are in centimetres. Work out the area of the square. You must show your working.
[4 marks]
Not drawn accurately
Answer
D6
A school trip costs £180 for the coach and £12 for each pupil's ticket. The school charges each pupil £18. There are $n$ pupils on the trip.
(a)
Write an expression, in terms of $n$ , for the total cost of the trip in pounds.
[1 mark]
Answer
(b)
The money the school collects must be more than the total cost. Write an inequality and solve it.
[2 marks]
Answer $n \dots\dots$
(c)
Write down the smallest number of pupils that will do this.
[1 mark]
Answer
Total for D6: 4 marks
D7
Mina's pocket money goes up by $8%$ every year. This year she gets £12.50 a week. Work out how much a week she will get in 3 years' time. Give your answer to the nearest penny.
[3 marks]
Answer
D8
Phone plan A costs £12 a month plus 5p for each minute of calls. Phone plan B costs £18 a month plus 3p for each minute of calls.
(a)
Write an equation and solve it to find the number of minutes in a month for which the two plans cost the same.
[3 marks]
Answer
(b)
Jo uses 420 minutes of calls in a month. Which plan is cheaper for her, and by how much?
[2 marks]
AnswerPlan $\dots$ , by £ $\dots\dots$
Total for D8: 5 marks
D9
A circular flower bed has an area of 60 m².
(a)
Work out the diameter of the flower bed. Give your answer to 1 decimal place.
[3 marks]
Answer
(b)
Edging is put all the way round the flower bed. Edging is sold in whole metres and costs £4.50 a metre. Work out the cost of the edging.
[3 marks]
Answer
Total for D9: 6 marks
D10
Write down every integer $n$ that satisfies $- 7 \leq 2 n + 1 < 9$ .
[3 marks]
Answer

Challenge

UKMT level

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
In a box, the ratio of red counters to blue counters is $5 : 3$ . Ten red counters are taken out and two blue counters are put in. Now there are the same number of red and blue counters. How many counters were in the box at the start?
[4 marks]
Hint 1 · What to try
Call one part of the ratio $k$ . Then there are $5 k$ red and $3 k$ blue counters at the start.
Hint 2 · The first line
After the change there are $5 k - 10$ red and $3 k + 2$ blue, and these are equal.
Hint 3 · The full method
$5 k - 10 = 3 k + 2$ gives $2 k = 12$ , so $k = 6$ . At the start there were $30$ red and $18$ blue, which is $48$ counters.
Answer

E2
A shop raises the price of a jacket by $x %$ . Later it lowers the new price by $x %$ . The final price is $4%$ less than the price at the start. Work out $x$ .
[4 marks]
Hint 1 · What to try
Try a value. A rise of $10%$ and then a fall of $10%$ multiplies the price by $1.1 \times 0.9$ .
Hint 2 · The first line
$1.1 \times 0.9 = 0.99$ , which is $1%$ less. You want a multiplier of $0.96$ .
Hint 3 · The full method
A rise and fall of $x %$ multiplies by $(1 + \frac{x}{100}) (1 - \frac{x}{100}) = 1 - (\frac{x}{100})^{2}$ . This is $0.96$ when $(\frac{x}{100})^{2} = 0.04$ , so $\frac{x}{100} = 0.2$ and $x = 20$ . Check: $1.2 \times 0.8 = 0.96$ .
Answer $x = \dots\dots$
E3
A circle and a square have the same perimeter. The square has sides of 9 cm. Work out the area of the circle. Give your answer to 1 decimal place.
[3 marks]
Hint 1 · What to try
The circumference of the circle is the same as the perimeter of the square.
Hint 2 · The first line
The circumference is $36$ cm, so $2 π r = 36$ .
Hint 3 · The full method
$r = \frac{36}{2 π} = 5.729 \dots$ cm. The area is $π r^{2} = π \times 5.729 \dots^{2} = 103.1$ cm².
Answer
E4
Work out the sum of all the integers $n$ that satisfy both $2 (n - 3) < 5 n + 9$ and $4 n - 1 \leq n + 17$ .
[3 marks]
Hint 1 · What to try
Solve each inequality on its own. Watch the sign when the $n$ terms are collected.
Hint 2 · The first line
The first gives $n > - 5$ . The second gives $n \leq 6$ .
Hint 3 · The full method
The integers are $- 4, - 3, \dots, 5, 6$ . The numbers from $- 4$ to $4$ add to $0$ , so the sum is $5 + 6 = 11$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can share an amount in a ratio and decide which pack is better value.
I can use a multiplier for a percentage change, repeated changes and reverse percentages.
I can work out the area and circumference of a circle and the area of a trapezium.
I can solve equations and inequalities with the unknown on both sides.

Answers and mark scheme

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