Percent change is one of 2 percent formulas in the arithmetic section of this library, and it is used at middle school · high school level.
Why percent change works
The size of a change only makes sense next to where it started. A 5 dollar rise is huge on a 50 dollar item and tiny on a 500 dollar one. Dividing the change by the starting value turns it into a share of the original, and multiplying by 100 renames that share as a percent.
What each symbol means
new and original are measured in the same units.
Percent change: when it holds
$\text{original}\ne0$; a positive result is an increase and a negative result is a decrease.
When it stops applying
It has no answer when the original value is 0, since dividing by zero is not allowed, and there is no honest way to say what percent of nothing a rise represents. Growth from 0 to 12 gets described in plain numbers instead.
Percent change: a worked example
From $50$ to $62$: $\frac{62-50}{50}(100\%)=24\%$ increase.
The mistake to avoid
What people do: Students divide the change by the new number instead of the original one.
Why it goes wrong: The starting value is the reference point for the comparison, and using the ending value answers a different question.
Do this instead: Always put the original on the bottom. Going from 50 to 62, the change is 12 and the answer is 12/50 = 24 percent. Dividing by 62 gives 19.35 percent, which is the answer to a question nobody asked.
Percent change: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. new and original are measured in the same units.
- Check the conditions before substituting. $\text{original}\ne0$; a positive result is an increase and a negative result is a decrease.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most arithmetic slips.
Where this formula fits
- Subject
- Arithmetic formulas — 13 entries in this library
- Topic
- Percent
- Level
- Middle school · High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where percent change comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Understanding Percentages — the lesson behind this formula: translate among percentages, decimals, and proportions.
- Arithmetic Calculator — check your substitution and the value it produces.
- Study arithmetic — the subject guide that explains the ideas these formulas compress.
- Arithmetic Practice — questions that make you retrieve the formula instead of recognising it.
- All 13 arithmetic formulas — the full grouped reference, or the complete formula library.
Questions about percent change
Why does a rise and a fall of the same percent not cancel out?
Because they are taken from different starting points. Going from 50 up to 62 is a 24 percent rise, but dropping 24 percent from 62 lands you at 47.12, not back at 50.
What is the difference between percent and percentage points?
Points measure the plain gap between two percents. A rate moving from 4 percent to 6 percent rose by 2 percentage points, which is a 50 percent increase, and news stories mix these up constantly.
Can percent change be more than 100 percent?
For an increase, yes: doubling is a 100 percent rise and tripling is 200 percent. A decrease cannot pass 100 percent, though, since losing everything is as far down as you can go.
Why are the absolute value bars around the original?
They keep the sign of the answer honest when the starting value is negative, such as with a temperature or a debt. The sign should come from the direction of the change alone, not from the sign of where you began.