Percentage Change

Your score went from 50 to 75 — up 25 points. But "up 25 points" resists comparison: one student rose from 50 to 75, another from 90 to 100; both gained 25, yet the achievement is worlds apart. This is a job for percentages — to describe a change, you must first say what it is relative to.

Divide the change by the original

From 50 to 75: up 25, divided by the original 50. $25 \div 50 = 50\%$ — a 50% increase. Not 25%: the base is the original number, not the change itself.

Now the reverse, 75 to 50: down 25, and $25 \div 75 \approx 33\%$ — only about 33%. The same 25 points, yet up counts as 50% while down counts as 33%, because the two divisions use different bases.

The formula, and two quick accounts

Percent change = change ÷ original × 100%.

Monthly pay rises from 4000 to 4600: the change is 600, $600 \div 4000 = 15\%$ — a 15% raise;
A jacket drops from 200 to 150: the change is 50, $50 \div 200 = 25\%$ — a 25% discount.

InteractiveChange Lab

Discount %37

37% = 37/100 = 0.37 = 37/100

Every percent-change question ends at the same spot: the current amount is what percent of the original? Drag the grid to 80% — after a 20% discount, that is exactly what you pay. Drag it to 96% — and hold that number for the next section.

Successive changes do not add

Up 20% then down 20% feels like returning to the start. Compute instead: $100 \times 1.2 = 120$ , then $120 \times 0.8 = 96$ .

Rise and fall don't cancel

Up 20% then down 20% leaves 96% of the original — a net loss of 4%. Once you have risen, the 20% drop is taken on the bigger base of 120. Successive changes multiply: $1.2 \times 0.8 = 0.96$ . Likewise, +10% then +20% is not 30% but $1.1 \times 1.2 = 1.32$ — 32%.

This multiply-step-by-step move is the star of simple and compound interest.

Working backwards: from the result to the original

A tag reads "96 after 20% off". What was the original price? The discounted price is 80% of the original, so

$96 \div 0.8 = 120$

The original price was 120. Check: 80% of 120 is exactly 96. Correct.

Ask one question backwards first

For reverse problems, ask first: the number in my hand is what percent of the original? Then divide by that percent. Grabbing 96 and multiplying by 1.2 instead treats the discounted price as the base — the classic way to tumble.

Check yourself

Quick quiz

1. A price rises from 50 to 75. By what percent did it rise?
2. Up 20%, then down 20%. The result?
3. After a 20% discount a jacket costs 96. What was the original price?