Reverse Percentage

Statement

To find the original value before a percentage change:

\[ \text{Original} = \frac{\text{Final}}{1 + \tfrac{r}{100}} \]

Here, \(r\) is the percentage change. If it’s an increase, \(r>0\); if it’s a decrease, \(r<0\).

Why it’s true

• A percentage increase means: Final = Original × (1 + r/100).
• Rearrange to find Original = Final ÷ (1 + r/100).
• Similarly, for a percentage decrease: Final = Original × (1 - r/100).
• Rearranging works the same way.

Recipe (how to use it)

1. Identify the final amount and the percentage change \(r\).
2. Add/subtract the percentage in multiplier form: \(1 + r/100\).
3. Divide the final by this multiplier.
4. Simplify to find the original.

Spotting it

Look for wording like “after a 20% increase, the price is …” or “after a 15% reduction, the value is …”. These are reverse percentage problems.

Common pairings

• Price increases (shops, VAT, discounts).
• Population growth/decline questions.
• Financial exam problems with “before” and “after” values.

Mini examples

1. Final price £120 after 20% increase → Original = 120 ÷ 1.2 = £100.
2. Final £85 after 15% decrease → Original = 85 ÷ 0.85 = £100.

Pitfalls

• Adding/subtracting percentages instead of using multipliers.
• Forgetting that a decrease means dividing by less than 1.
• Mixing up original and final values.

Exam strategy

• Always write the multiplier first: (1 + r/100).
• Check whether it’s an increase or decrease.
• Double-check by recalculating the percentage change forwards.

Summary

Reverse percentages let us work backwards to the original value. Divide the final value by the multiplier (1 ± r/100) depending on whether it was an increase or decrease.