Percent increase and decrease

A price that goes up 20% and then down 20% ends 4% lower than where it started.

The 20% rise and the 20% sale at a snack bar

A made-up story. The owner of a small snack bar raises the price of a plate of tteokbokki from 5,000 won to 6,000 won, a 20% rise. A month later business is slow, so she puts up a sign: everything 20% off. Customers now pay 4,800 won. She has not gone back to 5,000. The rise was 20% of 5,000, which is 1,000 won. The cut was 20% of 6,000, which is 1,200 won. Same percent, different starting point, and 4,800 ÷ 5,000 = 0.96 leaves the price 4% below where it began.

A 25-cent rise on a 2.50 fare

A bus fare rises from 2.50 to 2.75 dollars. The fare went up by 25 cents, but whether that is a lot depends on what it is compared with. A percent increase answers that. OpenStax gives two steps: find the amount of increase, new amount minus original amount, then find what percent that increase is of the original amount. Here 2.75 − 2.50 = 0.25, and 0.25 ÷ 2.50 = 0.1, so the fare rose 10%.

A percent decrease is built the same way: amount of decrease is original minus new, divided by the original. A jar of coffee drops from 80 to 68 dollars: 80 − 68 = 12, and 12 ÷ 80 = 0.15, a 15% decrease. The word original matters in both. Divide by where you started, never by where you ended.

Why 20% up and 20% down lands on 96

Start at 100. A 20% increase gives 120. A 20% decrease from 120 takes away 24, not 20, because 20% of 120 is 24. You land at 96, which is 4% below where you began. In multipliers: 1.2 × 0.8 = 0.96. The same thing happens in the other order, 0.8 × 1.2 = 0.96. Equal percents up and down never cancel, and the loss grows with the size of the percent: up and down 50% gives 1.5 × 0.5 = 0.75.

A 10% rise is × 1.1

An increase of r percent multiplies by 1 + r/100; a decrease multiplies by 1 − r/100. A 10% rise is × 1.1, a 35% fall is × 0.65. To undo a change, divide by its multiplier. If a price is 150 after a 25% increase, it was 150 ÷ 1.25 = 120 before. Taking 25% off 150 would give 112.50, which is wrong for the same reason as the 20% example above.

Repeated changes multiply too. Something that grows 10% a year for three years is multiplied by 1.1 × 1.1 × 1.1 = 1.331, a total rise of 33.1%, not 30%. Each year the 10% is taken of a bigger number.

Prices can rise 150% but cannot fall 150%

An increase can be any size. Doubling is a 100% increase: 40 to 80 adds 40, which is 100% of 40. Going from 40 to 100 adds 60, which is 150% of 40. A decrease cannot go past 100%, because losing 100% leaves nothing. A sentence like "prices fell 150%" is a mistake; the writer probably meant that something else rose 150%.

From 25 to 20 is 20%; from 20 to 25 is 25%

A shirt is cut from 25 to 20 dollars. The cut is 5 dollars, and 5 ÷ 25 = 20%, so it is a 20% decrease. When the sale ends and the price goes back from 20 to 25, the rise is the same 5 dollars, but now the original is 20, and 5 ÷ 20 = 25%. Both statements are correct. The size of a percent change always depends on which number you start from, so a report of a percent change should say what it is measured against.