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Level 8Grade 8skill: percent_change· 8 min read

Percentage Increase and Decrease

In short: Turn every percentage into a multiplier: increase by p% means ×(1 + p/100), decrease means ×(1 − p/100). Multiply multipliers for changes in a row, and divide to reverse a change. The trap to remember: a rise then an equal-percent fall lands below the start (1.2 × 0.8 = 0.96), because the second change acts on a different base.

The multiplier method (one multiplication per change)

Most percentage-change mistakes come from doing it in two clumsy steps: work out the change, then remember to add or subtract it. The change gets computed and the second step gets forgotten. Kill that whole class of error by turning every percentage into a single multiplier.

To increase by 15%, you keep the original 100% and add 15%, so you keep 115% — multiply by 1.15. To decrease by 15%, you keep 85% — multiply by 0.85. One multiplication, done. No separate add-on step to forget.

The real power shows up with several changes in a row: you just multiply the multipliers together. A rise of 10% followed by a rise of 20% is 1.10 × 1.20 = 1.32, a 32% rise — not 30%. And to go backwards (find the original from the final), you divide by the multiplier instead of multiplying.

  1. Turn each percentage into a multiplier: increase by p% → ×(1 + p/100); decrease by p% → ×(1 − p/100).
  2. Apply changes in order by multiplying the multipliers together.
  3. To find a percentage change from two amounts: (new − old) ÷ old, then ×100.
  4. To reverse a change (final → original), divide by the multiplier; never take the percentage of the final amount.
  5. Check the base every time: each percentage acts on the value that exists at that moment, not on the starting value.

Why a rise then an equal fall never returns you to the start Why it works

Increase by p%
multiply by (1 + p/100)

£200 up 20% → 200 × 1.2 = £240

Decrease by p%
multiply by (1 − p/100)

£240 down 20% → 240 × 0.8 = £192

Two changes in a row
multiply the multipliers, don't add the percentages

up 25% then down 10% → 1.25 × 0.90 = 1.125 (a 12.5% rise)

Reverse a change
divide by the multiplier

£360 incl. 20% VAT → 360 ÷ 1.2 = £300

Here is the insight the exam keeps testing: after a 20% rise, the matching 20% fall acts on the larger amount — £240, not £200 — so it strips away more pounds than the rise ever added. The percentages are equal but their bases are not, so the two can never cancel and you land below where you started. That is why you must never add percentages or assume they quietly undo each other.

Worked examples

A share is worth £500. It rises 25% one week and falls 25% the next. What is it worth now? = £468.75 — a 6.25% fall, not back to £500.
  1. Rise 25%: multiplier 1.25 → 500 × 1.25 = £625.
  2. Fall 25%: multiplier 0.75 → 625 × 0.75 = £468.75.
  3. The fall acted on £625, not £500, so it removed more than the rise added.
  4. Net multiplier: 1.25 × 0.75 = 0.9375, a 6.25% loss.
Sophie's salary is £2,000. It rises 10%, and later rises another 20%. What is her new salary? = £2,640 (a 32% increase overall).
  1. Do not add the percentages to get 30% — that would give £2,600 and is wrong.
  2. First rise: 2000 × 1.10 = £2,200.
  3. Second rise acts on £2,200: 2200 × 1.20 = £2,640.
  4. Check with one multiplier: 1.10 × 1.20 = 1.32, a 32% rise.
A phone costs £360 including 20% VAT. What was the price before VAT? = £300.
  1. Adding VAT means the pre-VAT price was multiplied by 1.20 to reach £360.
  2. So going backwards you divide, not take 20% off: 360 ÷ 1.20 = £300.
  3. Check: 300 × 1.20 = £360. ✓
  4. Taking 20% off £360 gives £288, which is wrong — that uses the larger amount as the base.
A shop raises a £150 coat by 40%, then puts it in a sale at 30% off. Is the sale price above or below £150? = £147 — 2% below the original £150.
  1. Raise 40%: 150 × 1.40 = £210.
  2. Sale 30% off: 210 × 0.70 = £147.
  3. Combined multiplier: 1.40 × 0.70 = 0.98, a 2% fall overall.
  4. Adding +40% and −30% to guess +10% is wrong — percentages on different bases don't add.

Common mistakes (and how to avoid them)

Taking the percentage of the wrong amount

✗ A coat is £96 after a 20% discount. Original price = 96 + 20% of 96 = 96 + 19.20 = £115.20.

The 20% was taken off the original price, not the sale price. Adding 20% of £96 uses the wrong base entirely.

Fix: The sale price is 80% of the original: 96 = 0.8 × original, so original = 96 ÷ 0.8 = £120. Check: 120 × 0.8 = £96. ✓

base_confusion

Working out the change but not applying it

✗ Increase 45 by 30%: 30% of 45 = 13.5, so the answer is 13.5.

You found the size of the increase but stopped there. The question asks for the new value, not the change.

Fix: Add it back — or better, use the multiplier: 45 × 1.30 = 58.5. (13.5 is only the extra bit: 45 + 13.5 = 58.5.)

partial_computation

Using the percent number instead of computing the percent

✗ Increase 240 by 15%: 240 + 15 = 255.

You added the number 15 as if it were pounds. 15% of 240 is not 15 — the percent has to be worked out relative to the amount.

Fix: 15% of 240 = 36, so 240 + 36 = 276, i.e. 240 × 1.15 = 276.

percent_as_whole

Adding percentages instead of compounding them

✗ £2,000 rises 10% then rises 20%: that's 30%, so 2000 × 1.30 = £2,600.

The second 20% acts on the already-raised amount, not the original, so the percentages don't simply add.

Fix: Multiply the multipliers: 1.10 × 1.20 = 1.32. 2000 × 1.32 = £2,640.

wrong_operation

Practice this skill

Try adaptive percentage-change questions: successive rises and falls, discounts and VAT on the correct base, and reversing a change to find the original. Watch for the traps — equal up-then-down percentages never return you to the start.

Related skills

First, make sure of

Frequently asked questions

If a price goes up 20% and then down 20%, why isn't it back to where it started?

Because the two 20%s act on different amounts. Starting at £200, a 20% rise adds £40 to reach £240. The 20% fall then acts on £240, removing £48 — more than was added — leaving £192. As multipliers: 1.2 × 0.8 = 0.96, a 4% overall loss. Equal percentages on unequal bases never cancel.

Can I just add two percentage changes together?

No. Adding works only by luck. For a 10% rise then a 20% rise, adding gives 30%, but the true result is 1.10 × 1.20 = 1.32, a 32% rise, because the second change acts on the already-increased amount. Always multiply the multipliers instead of adding the percentages.

How do I find the original price from a sale price or a VAT-inclusive price?

Divide by the multiplier, don't take a percentage off the final amount. If £360 includes 20% VAT, the pre-VAT price is 360 ÷ 1.20 = £300. If a coat is £96 after 20% off, the original is 96 ÷ 0.8 = £120. Taking 20% off the final figure uses the wrong base and gives the wrong answer.

What's the quickest way to find a percentage increase or decrease between two numbers?

Use (new − old) ÷ old × 100. A population going from 250 to 320 changes by (320 − 250) ÷ 250 = 70 ÷ 250 = 0.28 = a 28% increase. The key is dividing by the original value, not the new one.

Does the order of a discount and a tax matter?

For the final price, multiplying by 0.75 (25% off) and 1.20 (20% tax) gives the same answer either way, because multiplication is commutative: 0.75 × 1.20 = 0.90. But each percentage must still be applied to the correct running amount, and questions about the intermediate values do depend on order, so read carefully.