Percent change with multipliers
Most percent mistakes come from treating a percentage as an amount to add or subtract. Turn each change into a single multiplier instead, and the arithmetic becomes one multiplication per change, in any order, with nothing to remember about which base you are using.
Every change is a multiplier
An increase of 25 percent multiplies by 1.25. A decrease of 20 percent multiplies by 0.80. The multiplier is 1 plus the rate for a rise, and 1 minus the rate for a fall. Once a change is written this way, the percentage disappears and what remains is ordinary multiplication, which is why the method survives numbers that do not divide neatly.
- Up 8 percent: multiply by 1.08
- Down 15 percent: multiply by 0.85
- Up 100 percent: multiply by 2
- Down 100 percent: multiply by 0
Changes in a row multiply
Two changes in sequence multiply their multipliers, in either order. A price up 25 percent and then down 20 percent is 1.25 times 0.80, which is 1.00, so the price ends exactly where it started. The same two changes written as plus 25 and minus 20 tempt you to answer plus 5, and that is the trap the question is built around.
A percent of a percent
When a question takes a percentage of a group and then a percentage of that smaller group, the two rates multiply. If 60 percent of students own a phone and 40 percent of those also own a tablet, the share with both is 0.60 times 0.40, which is 24 percent of all students. Adding the rates would give 100 percent, which should feel wrong immediately.
Working backwards
If the final value equals the original times a multiplier, then the original equals the final divided by that multiplier. A coat that costs 60 dollars after a 25 percent discount was 60 divided by 0.75, which is 80 dollars. Dividing by the multiplier is the step that gets skipped, and it is the whole question when the original is what is being asked.
When the question asks for percent change
Percent change is the difference divided by the original, times 100. The original is the denominator, always, not the new value. A jump from 40 to 50 is 10 divided by 40, which is a 25 percent increase, while the same jump read backwards, from 50 down to 40, is 10 divided by 50, which is a 20 percent decrease. Same numbers, different answers, because the base changed.
Why don't a 25 percent increase and a 20 percent decrease cancel out?
They do cancel, and that is the point. 1.25 times 0.80 is 1.00. What does not cancel is a rise and a fall of the same percentage: up 25 percent then down 25 percent multiplies to 0.9375, a net loss of 6.25 percent.
How do I increase a number by 8 percent?
Multiply by 1.08. For a compound increase of 8 percent a year for three years, multiply by 1.08 three times, which is 1.08 cubed, not 1.24.
The price rose 20 percent to 60 dollars. What was it before?
Divide by 1.20, which gives 50 dollars. Subtracting 20 percent of 60 would give 48, and that is the wrong answer the question is waiting for.
What is the difference between percent change and percentage points?
Percentage points measure a difference between two percentages, so 45 percent to 50 percent is 5 percentage points. Percent change measures the difference relative to the starting value, which here is 5 divided by 45, about 11 percent.
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