Percentage change: the 20% trap

The backpack that does not quite bounce back

A 20% decrease followed by a 20% increase leaves 96% of a positive starting amount. The percentages use different bases, so they do not cancel. Cambridge’s NRICH explores this changing-base puzzle.

Imagine a backpack priced at $100. A 20% reduction removes $20, leaving $80. Later, its price rises by 20%. Surely the original price returns? The backpack has other plans.

The second 20% belongs to 80

The increase uses the current $80 price. Ten percent of 80 is 8, so 20% is 16. Add that: 80 + 16 = 96.

You removed 20 but added only 16. As in our percentage basics lesson, a percentage is a share of something. Here, the “something” changed between steps. Twenty percent of 100 and twenty percent of 80 are different amounts.

How do you calculate percentage change?

For a positive original amount, calculate (new amount − original amount) ÷ original amount × 100%. A positive result is an increase; a negative result is a decrease. OpenStax explains comparing the change with the original amount.

From 100 to 80, the change is −20: −20 ÷ 100 × 100% = −20%. For the next step, 80 becomes the original amount. From 80 to 96, the change is 16: 16 ÷ 80 × 100% = 20%.

What increase would really get you back?

To get from 80 back to 100, you need an increase of 20. Compare that increase with 80: 20 ÷ 80 × 100% = 25%.

So a 20% drop needs a 25% rise to reverse it. Our fictional backpack does return to $100—just with a different percentage on the return journey.

Before combining percentages, write down each starting amount. The two percentage signs may match perfectly while quietly referring to different wholes.