Finding an original amount | Year 8

01 / LEARN

The idea, step by step

Work backwards from the final value

If a sale takes 20% off, the sale price is 80% of the original. The final price is not the whole used to calculate the discount. Write the relationship before calculating: original amount multiplied by 0.8 equals the sale amount.

To undo multiplication, divide by the multiplier. If a value has increased by 30%, its final amount is 130% of the original, so divide by 1.3. A discount reverses to a larger original; an increase reverses to a smaller original.

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

Check a reverse calculation

Once you find a possible original, apply the original percentage change to it. This checks both the arithmetic and your choice of multiplier. For money, keep enough precision before rounding to cents.

For two successive changes, multiply their multipliers to form the overall multiplier. You may then divide the final amount by that product. Keep the changes in their stated order when explaining the calculation.

A common trap

To undo a 20% discount, subtract 20% from the sale price.

Instead: Undo the multiplier 0.8 by division. A sale price of 96 AUD gives $96 \div 0.8 = 120$ AUD.

02 / WORKED EXAMPLES

See the thinking

Example 1

Reverse a discount

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

The bag now costs 80% of its original price.
$original = 72 \div 0.8 = 90$ .
Check: 20% of 90 is 18 and $90 - 18 = 72$ .

90 AUD

Example 2

Reverse an increase

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

The new number is 110% of the original.
$264 \div 1.1 = 240$ .
Check: $240 + 24 = 264$ .

240 students

Check your understanding

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

Show answer

The multiplier is 1.2, so $84 \div 1.2 = 70$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF 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 ↗

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