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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.

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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

  1. A 35% reduction means the sale price is 65% of the original.
  2. Multiplier = 1 − 0.35 = 0.65
  3. Original = 78 ÷ 0.65
  4. Original = £120
Answer£120
Example 2

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

Working

  1. 20% VAT means the price with VAT is 120% of the original.
  2. Multiplier = 1.20
  3. Original = 564 ÷ 1.20
  4. 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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