Finding an original amount | Year 8

01 / LEARN

Understand the idea.

Work backwards from the final value

A reverse percentage problem gives the final amount and asks for the original whole. After a $20 %$ discount, the final price is $80 %$ of the original. Write that relationship before calculating: $0.8 \times original = final$ .

Undo multiplication by dividing by the multiplier. After a $30 %$ increase, divide by $1.3$ ; after a $20 %$ discount, divide by $0.8$ . Reversing a discount should produce a larger original, while reversing an increase produces a smaller one.

original amount = \frac{final amount}{change multiplier}

Check a reverse calculation

Check your proposed original by applying the stated percentage change forwards. This tests both the calculation and the chosen whole. Adding the discount percentage to the sale price does not reverse a discount, because that percentage would use the wrong base.

For successive percentage changes, multiply their multipliers to obtain the overall multiplier, then divide the final value by that product. Keep sufficient digits throughout a reverse calculation and round money only when the context requires cents.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Reverse a discount

A bag costs 72 AUD after a 20% discount. Find its original price.

Worked solution

1
Identify the relationship to the original.
$100 % - 20 % = 80 % = 0.8$
The sale price is eighty percent of the original price.
2
Undo the discount multiplier.
$original price = 72 \div 0.8 = 90 AUD$
Dividing by zero point eight reverses the multiplication that produced the sale price.
3
Check the original price forwards.
$\begin{aligned} 90 \times 0.20 & = 18 AUD \\ 90 - 18 & = 72 AUD \end{aligned}$
Removing twenty percent of ninety reproduces the stated sale price.

Answer

90 AUD

EXAMPLE 02

Reverse an increase

A school has 264 students after enrolment increases by 10%. How many students were there before?

Worked solution

1
Express the increased total as a multiplier.
$100 % + 10 % = 110 % = 1.1$
The new enrolment includes all the original students plus ten percent more.
2
Recover the earlier enrolment.
$264 \div 1.1 = 240 students$
Dividing by the growth multiplier restores the original whole.
3
Verify the increase.
$\begin{aligned} 240 \times 0.10 & = 24 students \\ 240 + 24 & = 264 students \end{aligned}$
Adding ten percent of the proposed original reproduces the given total.

Answer

240 students

03 / YOUR TURN

Try it yourself.

Check your understanding

A number becomes 84 after increasing by 20%. What was it?

Show answer & reasoning

After a $20 %$ increase the multiplier is $1.2$ . The original is $84 \div 1.2 = 70$ ; checking gives $70 + 0.2 \times 70 = 84$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

A price is 64 AUD after a 20% discount. Find the original price.
2 marks
An amount becomes 117 after a 30% increase. Find its original value.
2 marks
A ticket costs 42 AUD after a 30% reduction. Find its original price.
2 marks
A town population becomes 168 after a 40% increase. Find the original population.
2 marks
A student says a sale price of 96 AUD after 20% off means the original was $96 - 19.20 = 76.80$ AUD. Explain the error and find the correct original price.
3 marks
A jacket is discounted by 25%, then its reduced price is discounted by a further 10%. It finally costs 81 AUD. Find the original price.
3 marks

Read the worked solutions

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

A price is 64 AUD after a 20% discount. Find the original price.
2 marks
$64 \div 0.8 = 80$ .
Check: $80 \times 0.8 = 64$ .
80 AUD
An amount becomes 117 after a 30% increase. Find its original value.
2 marks
$117 \div 1.3 = 90$ .
Check: $90 \times 1.3 = 117$ .
90
A ticket costs 42 AUD after a 30% reduction. Find its original price.
2 marks
The remaining fraction is 0.7.
$42 \div 0.7 = 60$ .
60 AUD
A town population becomes 168 after a 40% increase. Find the original population.
2 marks
$168 \div 1.4 = 120$ .
Check: an increase of 48 is 40% of 120.
120
A student says a sale price of 96 AUD after 20% off means the original was $96 - 19.20 = 76.80$ AUD. Explain the error and find the correct original price.
3 marks
The sale price represents 80% of the original.
$96 \div 0.8 = 120$ .
A discount must make the final price lower, so an original below 96 AUD is impossible.
The original is 120 AUD; the student applied a new discount rather than reversing it.
A jacket is discounted by 25%, then its reduced price is discounted by a further 10%. It finally costs 81 AUD. Find the original price.
3 marks
The overall multiplier is $0.75 \times 0.9 = 0.675$ .
$81 \div 0.675 = 120$ .
Check: $120 \times 0.75 = 90$ and $90 \times 0.9 = 81$ .
120 AUD

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Percentage increase/decrease; finding an original amount; discounts and mark-ups; percentage comparisons; budgets and purchase decisions; modelling and checking financial problems with digital tools. Simple interest can be offered here as a school-dependent application and developed in Year 9.

Official curriculum source ↗

In this lesson

Understand the idea.

Work backwards from the final value

Check a reverse calculation

See the method in action.

Reverse a discount

Identify the relationship to the original.

Undo the discount multiplier.

Check the original price forwards.

Reverse an increase

Express the increased total as a multiplier.

Recover the earlier enrolment.

Verify the increase.

Try it yourself.

Check your understanding

Keep practising