The salary of a person was reduced by 40%. By what percentage should the…

Concept: A percentage change always describes a movement measured against the base it acts on, and that base is whatever the quantity is worth at the moment the change is applied. So when a quantity first falls and then has to be brought back to its starting value, the fall is measured on the original base while the restoring rise is measured on the smaller value that survived the fall. The restoring rise is therefore (shortfall ÷ value after the fall) × 100, and a fall of x% needs a rise of 100x/(100 − x)% to return to the start.

Applying it here:

1. Let the original salary be 100 units. Any base works, because the answer is a ratio and the units cancel.

2. The salary falls by 40%, and that 40% is taken on the original base of 100, so the fall is 40 units.

3. Salary after the fall = 100 − 40 = 60 units. This 60 is now the base for anything applied to the reduced salary.

4. Shortfall still to be made up = 100 − 60 = 40 units.

5. Express that shortfall on the new base: required rise = (40 ÷ 60) × 100 = (2/3) × 100 = 66 2/3%.

Cross-check: Raise 60 by 66 2/3%: the rise is (2/3) × 60 = 40, and 60 + 40 = 100, which is the original salary. The general formula agrees: 100x/(100 − x) with x = 40 gives 4000/60 = 66 2/3%.

Common slip: Adding the same 40% back on the reduced salary gives 60 + 24 = 84 units, short of 100, because 40% of 60 is smaller than 40% of 100. The percentage that comes off and the percentage that must go back on are never equal once the base has changed.