# Up 25%, then down 25%: visual puzzle and solution | SwaVid

A quantity increases by 25% and then decreases by 25%. Does it return to its starting value? If not, what is the overall percentage change?

Canonical: https://www.swavid.com/puzzles/quiz-4

Source: https://www.swavid.com/puzzles/quiz-4

# Up 25%, then down 25%

## No. It finishes 6.25% below its starting value.

### Choose an easy starting value

### Apply the 25% increase

### Apply the 25% decrease to the new value

### Compare the finish with the start

SwaVid visual puzzle

A quantity increases by 25% and then decreases by 25%. Does it return to its starting value? If not, what is the overall percentage change?

Detailed solution

The two 25% changes use different bases. The decrease is 25% of the larger, increased amount.

Use 100 units. A percentage result does not depend on the unit, and 100 makes each step easy to see.

Twenty-five percent of 100 is 25, so the increased quantity is 125.

The decrease is now 25% of 125, not 25% of the original 100. That decrease is 31.25.

The quantity moved from 100 to 93.75, a loss of 6.25 units out of the original 100.

Adding +25% and −25% gives zero only when both percentages are taken from the same base. Here the second base is 125, so the changes do not cancel.

- 1 Choose an easy starting value Use 100 units. A percentage result does not depend on the unit, and 100 makes each step easy to see. Start = 100
- 2 Apply the 25% increase Twenty-five percent of 100 is 25, so the increased quantity is 125. 100 × 1.25 = 125
- 3 Apply the 25% decrease to the new value The decrease is now 25% of 125, not 25% of the original 100. That decrease is 31.25. 125 × 0.75 = 93.75
- 4 Compare the finish with the start The quantity moved from 100 to 93.75, a loss of 6.25 units out of the original 100. 100 − 93.75 = 6.25, so the net change is −6.25%

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