Percent change: raises, discounts and points
Three everyday percentage questions, and the mistake each one invites.
A raise: from $52,000 to $55,120
Percent change is the difference divided by the starting value: (55,120 − 52,000) ÷ 52,000 × 100 = 6%. Divide by the old figure, not the new one; dividing by 55,120 would give 5.66%.
If prices rose 3.5% over the same year, the raise in real terms is about 2.4% (1.06 ÷ 1.035 − 1), not 2.5%.
A discount and its undo
A $80 jacket marked down to $60 is 25% off: 20 ÷ 80. Going back up from $60 to $80 is a 33.3% increase: 20 ÷ 60. The same $20 is a different percentage of a different base.
That is also why 10% up and then 10% down does not return to the start: 100 becomes 110, then 99.
To undo a change, divide by it rather than subtracting: the fall that cancels a 20% rise is 1 ÷ 1.2 − 1 = −16.7%, and the rise that cancels a 20% fall is 1 ÷ 0.8 − 1 = +25%.
Percent versus percentage points
A savings rate going from 4% to 5% rose by one percentage point, but by 25% of itself. News reports that say a rate "rose 1%" usually mean one point. When both readings matter, say both: up 1 point, a 25% increase.
Check your own
The percentage calculator has a mode for each of these questions and shows the formula with your numbers in it.