Percentage Increase and Decrease

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1. 1

   Turn the percentage into a multiplier

   15% increase → × 1.15 · 15% decrease → × 0.85

   The multiplier does in one step what most students do in two — find the percentage, then add or subtract it. Beyond being faster, it is the only practical way to handle repeated change and reverse percentages, so it is worth adopting early.

2. 2

   Apply it once

   £240 after a 15% rise → 240 × 1.15 = £276

   One multiplication gives the new amount directly. There is no intermediate step to lose track of, which removes the commonest error in the two-step method: computing the percentage correctly and then forgetting to add it on.

3. 3

   For percentage change, divide by the original

   £240 → £276. Change 36. 36 ÷ 240 × 100 = 15%

   The denominator is always the original amount, not the new one and not the difference. Dividing by the wrong figure is the defining error of this topic, and it produces an answer close enough to look right.

4. 4

   Repeated change means repeated multipliers

   Two years at 5% → × 1.05 × 1.05 = × 1.05² = × 1.1025

   Compound change multiplies; it does not add. Two years of 5% growth is 10.25%, not 10%. This is why the multiplier method matters — the two-step approach makes compound questions nearly unmanageable.

5. 5

   Reverse a percentage by dividing

   £276 is after a 15% rise. Original = 276 ÷ 1.15 = £240

   To undo a change, divide by the multiplier that caused it. Subtracting 15% from £276 gives £234.60, which is wrong — because that 15% would be taken from the wrong base. Reverse percentage questions are designed to catch exactly this.

Why the multiplier is worth the switch

Students taught to find 15% and then add it produce correct answers to simple questions and get stuck on everything beyond. Compound interest, depreciation, repeated growth and reverse percentages are all routine with multipliers and awkward or impossible without them.

The conversion is worth drilling until automatic: 8% increase is 1.08, 8% decrease is 0.92, 40% decrease is 0.6, and a 100% increase is 2. Students who hesitate here hesitate on every question in the topic.

• Increase by p% → × (1 + p/100)
• Decrease by p% → × (1 − p/100)
• Percentage change = change ÷ original × 100
• Repeated change → multiply the multipliers
• Reverse a change → divide by the multiplier

Reverse percentages, and why they catch people

A price is £276 after a 15% rise — what was it before? The instinct is to take 15% off £276, and it is wrong, because the original 15% was calculated from the smaller starting figure, not from £276. Taking 15% off gives £234.60 rather than £240.

The reliable method is to name the multiplier and divide by it. £276 is 1.15 times the original, so the original is 276 ÷ 1.15. Sale-price questions are the most common form of this and are deliberately written so that the wrong method gives a tidy-looking answer.

A rise and an equal fall do not cancel

A price rises 20% then falls 20%. It has not returned to where it started: 1.2 × 0.8 = 0.96, so it is 4% below the original. The fall is taken from a larger amount than the rise was added to, and the multipliers make this immediately visible.

This appears constantly in exams and in real reporting about wages and prices. It is a good example of a case where the multiplier method does not just save time but prevents a conclusion that intuition would otherwise get confidently wrong.

How we teach percentage change

We teach multipliers first and never teach the find-then-add method as a stepping stone. It is not a simpler version of the same idea — it is a different technique that has to be abandoned later, and students are reluctant to abandon something that has been working.

We drill the reverse percentage separately, because recognising it is most of the difficulty. Once a student reliably spots that a question gives the amount after a change and asks for the amount before, the method is a single division.