Percentage change tells you how much something grew or shrank relative to where it started. Prices, salaries, test scores, website traffic, weight, rent: almost anything that changes over time can be described with a percentage change. The formula is short, but there are a few traps. This guide covers the formula, worked examples, reverse calculations and spreadsheet versions.
The formula
- Percentage change = (new value − old value) ÷ old value × 100
If the result is positive, it is a percent increase. If it is negative, it is a percent decrease.
The key detail is the denominator: you always divide by the old (starting) value. Percentage change answers "how big is this change compared with where we started?"
Example: a percent increase
A monthly bill goes from $40 to $50.
- Change = 50 − 40 = 10
- 10 ÷ 40 = 0.25
- 0.25 × 100 = 25% increase
Example: a percent decrease
A jacket drops from $80 to $60.
- Change = 60 − 80 = −20
- −20 ÷ 80 = −0.25
- −0.25 × 100 = −25%, a 25% decrease
Increases and decreases are not symmetric
This is the single most common source of confusion. Going from 40 to 50 is a 25% increase, but going back from 50 to 40 is a 20% decrease, because the starting value is now 50:
- (40 − 50) ÷ 50 × 100 = −20%
The same absolute change (10) produces different percentages depending on the starting point. This is also why a 50% loss followed by a 50% gain does not get you back to where you started:
- Start at 100, lose 50% → 50
- Gain 50% of 50 → 75
To recover from a 50% drop, you need a 100% gain. Keep this in mind whenever you see headlines about prices or investments "falling and then recovering".
Percentage change vs percentage points
When the values you are comparing are already percentages, be careful with wording. If an interest rate moves from 4% to 5%:
- It rose by 1 percentage point (5 − 4).
- It rose by 25 percent in relative terms ((5 − 4) ÷ 4 × 100).
Both are correct, but they mean different things. Our post on percentage points vs percent covers this in depth.
Reverse calculations
Often you know the percentage change and need one of the values.
Find the new value after a known change:
- New = old × (1 + change ÷ 100)
- A $1,200 rent rises 5%: 1,200 × 1.05 = $1,260
- A $90 price falls 30%: 90 × 0.70 = $63
Find the original value before a known change:
- Old = new ÷ (1 + change ÷ 100)
- A price after a 20% increase is $72: 72 ÷ 1.20 = $60
- A price after a 25% discount is $45: 45 ÷ 0.75 = $60
A frequent error is to subtract the percentage from the new value instead of dividing. Taking 20% off $72 gives $57.60, not the original $60, because the 20% was originally applied to a smaller number. For more on discounts, see the discount guide.
Multiple changes in a row
When changes stack, multiply the factors rather than adding the percentages.
- A price rises 10% and then another 10%: 1.10 × 1.10 = 1.21, a 21% total increase, not 20%.
- A price falls 20% and then rises 20%: 0.80 × 1.20 = 0.96, a 4% total decrease.
This also applies to stacked discounts: "20% off, then an extra 10% off" is 0.80 × 0.90 = 0.72, or 28% off, not 30%.
Average percentage change over time
If something grows over several periods, a simple average of the yearly percentages can mislead you. The more accurate measure is the compound annual growth rate (CAGR):
- CAGR = (ending value ÷ starting value)^(1 ÷ number of periods) − 1
Example: from 1,000 to 1,331 over 3 years: (1,331 ÷ 1,000)^(1/3) − 1 = 1.1 − 1 = 0.10, or 10% per year.
When the starting value is zero or negative
- Starting value of zero: percentage change is undefined, because you cannot divide by zero. Describe the change in absolute terms instead ("went from 0 to 25 sign-ups").
- Negative starting values: common in finance (such as a loss turning into a profit). The standard formula can give confusing signs. Many analysts divide by the absolute value of the old number, or simply report the absolute change. State which method you used.
Percentage change in Excel and Google Sheets
If the old value is in A2 and the new value is in B2:
=(B2-A2)/A2 and format the cell as a percentage- Or
=B2/A2-1, which gives the same result
To avoid errors when A2 is zero, wrap it: =IF(A2=0,"n/a",(B2-A2)/A2).
Percentage change vs percentage difference
Percentage change has a clear before and after, so you divide by the old value. Percentage difference compares two values with no natural starting point (two stores' prices, two measurements). It divides by their average:
- Percentage difference = |A − B| ÷ ((A + B) ÷ 2) × 100
Use change for "over time" and difference for "side by side".
Quick mental-math tips
- 10% of anything: move the decimal one place left.
- 5%: half of 10%.
- 1%: move the decimal two places left.
- To estimate a change, find 10% of the old value and see how many "10%s" fit in the difference.
For instant answers, use the Percentage Lab calculator or its dedicated percent increase page.
Frequently asked questions
What is the formula for percentage change?
Percentage change equals the new value minus the old value, divided by the old value, multiplied by 100.
How do I know if it is an increase or a decrease?
A positive result is a percent increase and a negative result is a percent decrease.
Why is going from 50 to 40 a 20% decrease but 40 to 50 a 25% increase?
Because percentage change always divides by the starting value, and the starting value is different in each direction.
How do I find the original price before a percentage increase?
Divide the new price by one plus the percentage as a decimal. For example, after a 20% increase, divide by 1.20.
Can I add percentage changes together?
Not accurately. For consecutive changes, multiply the growth factors, such as 1.10 × 1.10 for two 10% increases.
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