Percentages • Topic 2 of 5

Percentage Change

Percentage change always uses the ORIGINAL value as the base: change% = (new − old)/old × 100. The classic GRE trap is the asymmetry of increase and decrease — a price that rises 25% and then falls 25% does NOT return to the start; it ends lower, because the second cut is taken on a larger number. The clean tool is the multiplier: +x% means × (1 + x/100), −x% means × (1 − x/100). Master the inverse phrasing too: if A is 25% more than B, then B is only 20% less than A, because the base switched from B to A — the shortcut is B is [x/(100 ± x)] × 100%. This idea shows up constantly in Quantitative Comparison: if you are told A is 20% more than B, you can immediately compare quantities without ever finding the actual numbers.

✅ Solved examples

1. A salary rises from $40,000 to $46,000. Find the percent increase.
Change = 6000 on a base of 40000. (6000/40000) × 100 = 15%.
2. A number is increased by 25% then decreased by 20%. Net change?
Multiplier = 1.25 × 0.80 = 1.00 ⇒ 0% net change (the classic cancelling pair).
3. If A is 40% more than B, by what percent is B less than A?
B is [40/(100+40)] × 100 = 4000/140 ≈ 28.6% less than A.
4. A’s income is 25% more than B’s; B’s is 20% more than C’s. A is what percent more than C?
A = 1.25B, B = 1.20C ⇒ A = 1.25 × 1.20 × C = 1.5C ⇒ 50% more.

✏️ Practice — try these, take hints as needed

1. A quantity rises from 80 to 100. Percent increase?
Base is 80.
Change = 20.
(20/80) × 100.
25%
2. A price falls from 100 to 80. Percent decrease?
Base is 100.
Change = 20.
(20/100) × 100.
20% (not 25%)
3. Increase 50 by 30%, then decrease the result by 30%. Final value?
× 1.3 then × 0.7.
Net multiplier 0.91.
50 × 0.91.
45.5
4. If A is 20% less than B, by what percent is B more than A?
Use x/(100 − x).
20/(100 − 20).
(20/80) × 100.
25%
5. X is 25% more than Y; Y is 25% less than Z. X is what percent of Z?
X = 1.25Y, Y = 0.75Z.
X = 1.25 × 0.75 × Z.
= 0.9375Z.
93.75%

📝 Topic test — 8 questions

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