What is important to know about Reverse percentages?

If a price is increased by 20%, the new price = original × 1.2, so original = new price ÷ 1.2.

What is important to know about Reverse percentages?

If a price is decreased by 30%, the new price = original × 0.7, so original = new price ÷ 0.7.

What is important to know about Reverse percentages?

The multiplier for an x% increase is (1 + x/100); for an x% decrease it is (1 − x/100).

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Number · Foundation & Higher

Reverse percentages

Reverse percentages find the original value before a percentage change was applied. You divide by the multiplier rather than multiplying by it.

Revising reverse percentages the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Reverse percentages GCSE” on the site for the complete guide.

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Key facts to remember

1If a price is increased by 20%, the new price = original × 1.2, so original = new price ÷ 1.2.
2If a price is decreased by 30%, the new price = original × 0.7, so original = new price ÷ 0.7.
3The multiplier for an x% increase is (1 + x/100); for an x% decrease it is (1 − x/100).
4Never find x% of the new value — always divide by the multiplier.
5These questions often use phrases like "after VAT", "after a discount", or "following a rise".

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Formulas

Original value (after increase)

Original = New value ÷ (1 + r/100)

r is the percentage increase

Original value (after decrease)

Original = New value ÷ (1 − r/100)

r is the percentage decrease

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Worked examples

Example 1

A jacket costs £78 after a 35% reduction. Find the original price.

Working

A 35% reduction means the sale price is 65% of the original.
Multiplier = 1 − 0.35 = 0.65
Original = 78 ÷ 0.65
Original = £120

Answer£120

Example 2

A TV costs £564 including 20% VAT. Find the price before VAT.

Working

20% VAT means the price with VAT is 120% of the original.
Multiplier = 1.20
Original = 564 ÷ 1.20
Original = £470

Answer£470

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Common mistakes

✗Finding x% of the new value instead of dividing by the multiplier (e.g. finding 35% of £78 and subtracting).

✗Using the wrong multiplier — a 35% decrease uses 0.65, not 0.35.

✗Confusing reverse percentages with percentage change.

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Exam tips

✓Always identify the multiplier first, then divide — never work backwards by finding a percentage of the given amount.

✓Check your answer: apply the percentage change to your original value and see if you get the price given.

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