KS3 Maths · Year 8 · Chapter 1: Percentages · Lesson 4

Repeated percentage changes

Year 8About 70 minutesCalculator allowed49 marks

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Quick questions to start.

A1
Needed today
Write down the multiplier for an increase of $7%$ .
[1 mark]
Answer
A2
Needed today
Work out $1.2 \times 0.9$ .
[1 mark]
Answer
A3
Last lesson
After a $25%$ increase, a price is $£45$ . Work out the price before the increase.
[1 mark]
Answer
A4
Year 7
Work out $1. 5^{2}$ .
[1 mark]
Answer

Example, then your turn

Year 8

For changes one after another, multiply by each multiplier in turn. The product of the multipliers is the single multiplier for the whole change. Your teacher works through each example with you. Then try the one beside it.

Example 1

A plant is $40$ cm tall. Its height increases by $15%$ , and then by a further $10%$ . Work out its height now.

Hint The second

10%

is a percentage of the NEW height, so multiply by

1.15

and then by

1.1

B1
A house is worth $£220 000$ . Its value increases by $5%$ , and then by a further $8%$ . Work out its value now.
[2 marks]
Answer

Example 2

A shirt costs $£32$ . The price goes up by $40%$ . Later it is reduced by $15%$ . Work out (i) the final price, (ii) the overall percentage change.

Hint Multiply the two multipliers to get one multiplier for the whole change.

B2
A museum had $4500$ visitors in May. In June the number rose by $12%$ . In July it fell by $25%$ . Work out (i) the number of visitors in July, (ii) the overall percentage change from May to July.
[3 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 3

A sofa costs $£850$ . It is reduced by $20%$ , and then by a further $15%$ . Work out the sale price and the overall percentage reduction.

Hint The

15%

is taken off the reduced price, so the reductions do not add to

35%

B3
A coat is reduced by $10%$ , and then by a further $25%$ . Work out the overall percentage reduction.
[2 marks]
Answer

Example 4

Ben puts $£5000$ in a savings account. It pays $3%$ interest each year, and he leaves the interest in the account. How much is in the account after $4$ years?

Hint Each year multiplies the amount by

1.03

, so

4

years multiply it by

1.0 3^{4}

B4
A car costs $£12 400$ . It loses $15%$ of its value each year. Work out its value after $3$ years.
[2 marks]
Answer

Example 5

Show that an increase of $20%$ followed by an increase of $15%$ is an overall increase of $38%$ .

Hint Work with the multipliers. You do not need a starting amount.

B5
Show that a decrease of $20%$ followed by an increase of $10%$ is an overall decrease of $12%$ .
[2 marks]

Practice

Year 8

C1
A price goes up by $50%$ and then down by $50%$ . Is it back to the price it started at? Give the overall percentage change.
[2 marks]
Answer
C2
A game costs $£60$ . Shop A puts the price up by $20%$ and then down by $10%$ . Shop B puts it down by $10%$ and then up by $20%$ . Work out the final price in each shop.
[2 marks]
AnswerA $£ \dots\dots$ B $£ \dots\dots$

C3
The population of a town is $12000$ . It grows by $5%$ in each of the next two years, and then falls by $10%$ in the third year. Work out the population after the three years.
[2 marks]
Answer
C4
A jacket costs $£150$ . Shop P takes $35%$ off. Shop Q takes $20%$ off, and then a further $20%$ off the sale price. Which shop is cheaper, and by how much?
[3 marks]
AnswerShop $\dots$ by $£ \dots\dots$
C5
A machine costs $£20 000$ . It loses $18%$ of its value each year. After how many whole years is it first worth less than $£10 000$ ?
[2 marks]
Answer
C6
A price goes up by $10%$ and then by a further $20%$ . It is now $£594$ . Work out the price before the two increases.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Tom says, "A decrease of $15%$ followed by a decrease of $25%$ is a decrease of $40%$ ."
Tom is wrong. Explain why, and work out the overall percentage decrease.
[3 marks]
AnswerOverall decrease $\dots\dots %$

D2
Kaya puts $£3500$ in a savings account for $3$ years. She leaves the interest in the account. The table shows the interest rate in each year.
(a)
Work out how much is in the account at the end of the $3$ years.
[3 marks]
Answer
(b)
Work out the overall percentage increase, to $1$ decimal place.
[2 marks]
Answer
Total for D2: 5 marks
D3
Ali has $£6000$ to invest for $3$ years. Bank X pays simple interest of $4%$ a year. Bank Y pays compound interest of $3.9%$ a year. Which bank gives Ali more money at the end of the $3$ years, and by how much? You must show your working.
[4 marks]
AnswerBank $\dots$ by $£ \dots\dots$
D4
A van costs $£24 000$ . It loses $25%$ of its value in the first year, and then $12%$ of its value in each year after that. Work out the value of the van after $3$ years.
[3 marks]
Answer
D5
A shop takes $20%$ off the price of a lamp, and then adds VAT at $20%$ to the sale price. The lamp costs $£75$ before both changes. Mo says, "The $20%$ off and the $20%$ added cancel out, so I pay $£75$ ." Is Mo correct? You must show your working.
[2 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A price goes up by $25%$ . Then it goes down by $x %$ . Overall, the price has gone down by $10%$ . Find $x$ .
[4 marks]
Hint 1 · What to try
Start with a price of $£100$ . Where does it end up, and where is it after the $25%$ rise?
Hint 2 · The first line
It goes from $£125$ down to $£90$ . What percentage of $125$ is taken off?
Hint 3 · The full method
$125 - 90 = 35$ , and $35 \div 125 = 0.28$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out several percentage changes in a row with one multiplier.
I can find the overall percentage change after two changes.
I can work out compound interest and depreciation with a power.
I can explain why two percentage changes do not just add.

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