Repeated percentage change | Year 10

01 / LEARN

Understand the idea.

Use the new base each time

Convert each change into a multiplier before combining them. A $20 %$ discount retains $0.8$ of the price, and the next $10 %$ discount retains $0.9$ of that reduced price, giving overall factor $0.8 \times 0.9$ .

The combined percentage is measured relative to the original amount. If the overall retained factor is $b < 1$ , the overall discount is $(1 - b) \times 100 %$ . Adding the discount percentages ignores the changed base.

Reversing a factor

To reverse multiplication by a nonzero factor $b$ , divide by $b$ . Reversing a $20 %$ decrease needs factor $\frac{1}{0.8} = 1.25$ , a $25 %$ increase, because recovery starts from the smaller amount.

Equal percentage increases and decreases do not cancel. Their multiplication order gives the same final amount when these are the only changes, but inserting fixed fees, payments or other transactions can change that conclusion.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Two discounts

A 250-dollar item has a 20% discount followed by another 10% discount. Find the final price and overall discount.

Worked solution

1
Apply the first discount.
$250 (0.80) = 200 dollars$
A $20 %$ reduction keeps $80 %$ of the original price.
2
Apply the second discount to the reduced price.
$200 (0.90) = 180 dollars$
The second $10 %$ is based on $200$ dollars, not $250$ dollars.
3
Find the overall retained fraction.
$\frac{180}{250} = 0.72$
The final amount is compared with the original price.
4
Convert the total reduction into a percentage.
$(1 - 0.72) \times 100 % = 28 %$
Retaining $72 %$ means a combined discount of $28 %$ .

Answer

180 dollars; 28% discount.

EXAMPLE 02

Up then down

An amount rises by 10% then falls by 10%. Find its net percentage change.

Worked solution

1
Write a multiplier for each change.
$increase factor = 1.10, decrease factor = 0.90$
Each percentage acts on the amount current at its own step.
2
Multiply the factors.
$1.10 \times 0.90 = 0.99$
The second change applies to the increased amount.
3
Interpret relative to the original amount.
$(0.99 - 1) \times 100 % = - 1 %$
The final amount retains $99 %$ , so the net change is a $1 %$ decrease.

Answer

1% decrease.

03 / YOUR TURN

Try it yourself.

Check your understanding

What increase reverses a 25% decrease?

Show answer & reasoning

$33 \frac{1}{3} %$

A $25 %$ decrease leaves factor $0.75 = \frac{3}{4}$ . Its reverse factor is $\frac{4}{3}$ , so the required increase is $(\frac{4}{3} - 1) 100 % = 33 \frac{1}{3} %$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

A 400-dollar price rises by 5%, then falls by 5%. Find the final price.
2 marks
Combine increases of 10% and 20%.
2 marks
Combine decreases of 10% and 30%.
2 marks
A final price is 216 dollars after a 10% discount. Find the original price.
2 marks
What percentage increase reverses a 40% decrease?
3 marks
An 800-dollar amount rises 20%, falls 25%, then rises 10%. Find the result and net change.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A 400-dollar price rises by 5%, then falls by 5%. Find the final price.
2 marks
$400 (1.05) (0.95) = 399$ .
399 dollars
Combine increases of 10% and 20%.
2 marks
$1.1 \times 1.2 = 1.32$ .
32% increase
Combine decreases of 10% and 30%.
2 marks
$0.9 \times 0.7 = 0.63$ .
37% decrease
A final price is 216 dollars after a 10% discount. Find the original price.
2 marks
$\frac{216}{0.9} = 240$ .
240 dollars
What percentage increase reverses a 40% decrease?
3 marks
The reverse factor $\frac{1}{0.6} = \frac{5}{3}$ exceeds 1 by $\frac{2}{3}$ .
$66 \frac{2}{3} %$
An 800-dollar amount rises 20%, falls 25%, then rises 10%. Find the result and net change.
3 marks
$800 (1.2) (0.75) (1.1) = 792$ ; the combined factor is $0.99$ .
792 dollars; 1% decrease.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Simple versus compound growth; compound interest; depreciation; repeated percentage change; savings/loan scenarios with supplied models; changing assumptions and checking results; spreadsheets/digital models.

Official curriculum source ↗

In this lesson

Understand the idea.

Use the new base each time

Reversing a factor

See the method in action.

Two discounts

Apply the first discount.

Apply the second discount to the reduced price.

Find the overall retained fraction.

Convert the total reduction into a percentage.

Up then down

Write a multiplier for each change.

Multiply the factors.

Interpret relative to the original amount.

Try it yourself.

Check your understanding

Keep practising