SAT · GRE · GMAT · Beginner · ~8 min

Why +20% then −20% isn't zero

Chain percentage changes and watch where the money really goes.

guess first, then play →

PredictFirst, a guess

\$100

jacket goes up by 20%, then the new price goes down by 20%. What does it cost now?

ExploreBuild a chain of changes

Start
100
After change 1 (+20%, ×1.2)
120
After change 2 (−20%, ×0.8)
96

Starting value100

Change 1+20%

Change 2−20%

Each change multiplies:

1.2 \times 0.8 = 0.96

. So 100 becomes 96: a net change of −4%, not ±0%.

Reverse the order: the final value never changes — multiplying doesn't care about order.

ConceptUndoing a change

A change of+25%

To undo +25% you need −20%. In multipliers:

1.25 \times 0.8 = 1

Rises are undone by smaller falls, and falls need bigger rises to recover. That is the reverse-percentage idea: divide by the multiplier, never just subtract the percentage.

ConceptRepeating a change: compounding

Change each time+10%Number of times5

Compounding (the truth): 161.05

Adding the percentages (the trap): 150

Break itTry the extremes

Exam trapsWhere marks get lost

After a 20% discount a shirt costs

\$60

. What was the original price?

An interest rate rises from 4% to 5%. By what percent did the rate increase?

Quantity A: the result of +30% then −30% on 200. Quantity B: the result of −30% then +30% on 200.

Worked solutionAn exam-style question, step by step

A stock rose 25% in January and fell 20% in February. In March it rose by

p

%, ending 10% above its price at the start of January. Find

p