Ratio, proportion and rates of change · KS3

Percentage Change

"20% off", "prices up 5%", "a 150% increase". Every percentage change is a multiplier: one number that takes you from old to new in a single step. Learn to find it and every increase, decrease, reverse and repeated change becomes the same calculation.

National curriculum KS3: solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest in financial mathematics; use multipliers

1 · Percentage of an amount: the 100-bar

The original amount is 100%, whatever its actual value. 1% is one hundredth of it; 25% is a quarter; 150% is one-and-a-half times. Slide the percentage and watch the bar: the number changes, the idea doesn't.

mental shortcuts:

2 · Increase and decrease: one multiplier

To increase by 20%, you end up with 100% + 20% = 120% of the original. 120% as a decimal is 1.2, so multiply by 1.2. To decrease by 20%, you keep 80%: multiply by 0.8. One multiplication, no adding step, and it reverses cleanly. The bar shows the original (teal) and what's added or removed.

Two-step method
Multiplier method

Multipliers at a glance. Up 5% → × 1.05. Up 50% → × 1.5. Up 100% → × 2 (doubling). Down 5% → × 0.95. Down 35% → × 0.65. Down 100% → × 0: nothing left. A decrease can't go beyond 100%, but an increase can: "up 250%" is × 3.5.

3 · Finding the percentage change

A price went from £40 to £46. What percentage increase is that? Find the change (£6), then ask what percentage of the original it is: 6 ÷ 40 = 0.15 = 15%. Always divide by the original, never the new value. The same change counts as a different percentage depending on where you started.

percentage change = new − originaloriginal × 100

Up 25% then down 25% is not back where you started. £100 → £125 → £93.75. The 25% off is taken from a bigger number than the 25% on was added to. Multipliers show it instantly: 1.25 × 0.75 = 0.9375, a 6.25% loss overall. Same reason a shop's "50% off, then another 50% off" is 75% off, not 100%.

4 · Reverse percentages: finding the original

A jacket costs £54 in a 10%-off sale. What was the full price? The trap is to add 10% back on to £54 (giving £59.40, wrong). The £54 is 90% of the original, not 100% of it. So divide: £54 ÷ 0.9 = £60. In general, new ÷ multiplier = original. Divide by the multiplier you'd have multiplied by.

Wrong: add it back on
Right: divide by the multiplier

5 · Repeated changes

A population grows 3% a year for 5 years. Not 15%: each year's 3% is taken of a bigger number. Multiply by 1.03 five times, which is × 1.03⁵. This is where percentage change meets the index laws, and it's the engine of compound interest (next page).

6 · Where percentage change lives

Shopping

Sales, VAT (20%, × 1.2), tips, service charges. "Was £80, now £60" is a 25% reduction; the shop will call it "save 25%", never "pay 75%".

Pay and prices

A 4% pay rise with 6% inflation is a real-terms cut: 1.04 ÷ 1.06 ≈ 0.98. Comparing two percentages means multiplying or dividing their multipliers.

Science

Percentage error, percentage yield in chemistry, percentage change in mass in osmosis practicals: all (new − original) ÷ original × 100.

Statistics and news

"Crime up 50%" from 2 cases to 3 is true and nearly meaningless. Always ask: 50% of what? A percentage change hides the original.

Computing

Progress bars, battery percentages, CPU load. A file compressed to 40% of its size is a 60% reduction; the multiplier is 0.4.

Finance

Interest, depreciation (a car losing 15% a year: × 0.85ⁿ), investment returns. The next page does simple interest; GCSE does compound.

Worked examples

Increase 360 by 15%
Multiplier: 100% + 15% = 115% = 1.15.
Multiply: 360 × 1.15 = 414. (Two-step check: 10% is 36, 5% is 18, so 15% is 54; 360 + 54 = 414 ✓)
A phone drops from £320 to £272. Percentage decrease?
Change: 320 − 272 = 48.
As a fraction of the original: 48 ÷ 320 = 0.15 → 15% decrease.
After a 20% rise, a rent is £900 a month. What was it before?
£900 is 120% of the original, so the multiplier was 1.2.
Divide: 900 ÷ 1.2 = £750. Check: 750 × 1.2 = 900 ✓. (Taking 20% off £900 gives £720: wrong.)
Exam-style: a car worth £12 000 loses 15% of its value each year. What is it worth after 3 years?
Multiplier each year: 0.85.
Three years: 12 000 × 0.85³ = 12 000 × 0.614125 = £7369.50.
Not 12 000 × 0.55 = £6600 (taking 45% off once). The losses shrink each year because each is 15% of a smaller number.

Quick check

What you've learned

The original is always 100%. An increase of p% means a multiplier of 1 + p/100; a decrease, 1 − p/100. New = original × multiplier, so original = new ÷ multiplier, which is how reverse-percentage questions are solved without the add-it-back trap. Percentage change is (new − original) ÷ original × 100, always over the original. Repeated changes multiply their multipliers, so n changes of p% is a power, not n × p%.