To calculate percentage change, subtract the old value from the new value, divide the result by the old value, and multiply by 100. As a formula: percentage change = (new − old) ÷ old × 100. A positive answer means an increase and a negative answer means a decrease; for example, a price that rises from $64 to $72 has increased by 12.5%.
The percentage change formula
percentage change = (new value − old value) ÷ old value × 100
Three things to keep straight:
- The old value is the base. You always divide by the starting value, not the ending one.
- The sign tells the direction. Positive is an increase, negative is a decrease, zero is no change.
- Multiply by 100 at the end. Without it, you have a decimal (0.125) rather than a percent (12.5%).
How to calculate percentage change in three steps
- Find the difference: new − old.
- Divide by the old value.
- Multiply by 100 and add the % sign.
Example 1: a percentage increase
A monthly bus pass goes from $64 to $72.
- Difference: 72 − 64 = 8
- Divide: 8 ÷ 64 = 0.125
- Multiply: 0.125 × 100 = 12.5% increase
Example 2: a percentage decrease
A website's weekly visits drop from 2,400 to 1,800.
- Difference: 1,800 − 2,400 = −600
- Divide: −600 ÷ 2,400 = −0.25
- Multiply: −0.25 × 100 = −25%, a 25% decrease
Example 3: decimals
A seedling grows from 3.5 cm to 4.2 cm.
- Difference: 4.2 − 3.5 = 0.7
- Divide: 0.7 ÷ 3.5 = 0.2
- Multiply: 0.2 × 100 = 20% increase
Example 4: an answer that needs rounding
A class grows from 45 students to 52.
- Difference: 52 − 45 = 7
- Divide: 7 ÷ 45 = 0.1555…
- Multiply: 15.555… ≈ 15.56% increase (rounded to two decimal places)
Round only at the last step. Rounding 7 ÷ 45 to 0.16 first would give 16%, which is noticeably off.
Percentage increase and decrease formulas
Some textbooks split the formula in two so the answer is always positive:
- Percentage increase = (new − old) ÷ old × 100, used when the new value is larger
- Percentage decrease = (old − new) ÷ old × 100, used when the new value is smaller
For the website example, the decrease formula gives (2,400 − 1,800) ÷ 2,400 × 100 = 25%, and you'd write "a 25% decrease." It's the same calculation as the signed version, just reported without the minus sign.
Why going up and coming back down aren't the same percentage
Because the base changes, an increase and the decrease that undoes it have different percentages.
- From 50 to 75: (75 − 50) ÷ 50 × 100 = +50%
- From 75 back to 50: (50 − 75) ÷ 75 × 100 ≈ −33.33%
The same 25-unit change is half of 50 but only a third of 75. This is also why a 20% rise followed by a 20% fall doesn't return you to where you started:
- 100 increased by 20% → 100 × 1.20 = 120
- 120 decreased by 20% → 120 × 0.80 = 96
The net result is a 4% decrease (from 100 to 96). To combine successive changes, multiply the factors: 1.20 × 0.80 = 0.96. Two 10% increases in a row work the same way: 1.10 × 1.10 = 1.21, a 21% total increase, not 20%.
Percent change vs percentage points
When the values are already percentages, there are two different ways to describe the change, and mixing them up is one of the most common errors in reports and news stories.
- Percentage points = the plain difference between two percentages.
- Percent change = that difference relative to the starting percentage.
| Change | Percentage points | Percent change |
|---|---|---|
| 20% → 25% | +5 points | (25 − 20) ÷ 20 × 100 = +25% |
| 12% → 10% | −2 points | (10 − 12) ÷ 12 × 100 ≈ −16.67% |
Both are correct; they answer different questions. "Support rose 5 percentage points" and "support rose 25%" describe the same change, but readers will picture very different things if you use the wrong one. When you write about data, name the measure you mean. This matters whenever you use numbers as evidence, for example in an argumentative essay or a report.
Working backward: applying a percentage change
Sometimes you know the percent and need the new value. Turn the percent into a multiplier:
- Increase by p%: new = old × (1 + p ÷ 100)
- Decrease by p%: new = old × (1 − p ÷ 100)
Examples:
- 80 increased by 15%: 80 × 1.15 = 92
- 250 decreased by 12%: 250 × 0.88 = 220
To find the original value from the new one, divide instead. A jacket costs $68 after a 15% discount. The sale price is 85% of the original, so:
- Original = 68 ÷ 0.85 = $80
- Check: 80 × 0.85 = 68 ✓
A common mistake is to add 15% of $68 back on. That gives 68 × 1.15 = 78.20, which is wrong because the discount was taken from $80, not from $68.
Percent change vs percentage difference
Percent change is for before-and-after situations: one value came first. When you're comparing two values that have equal standing, like prices at two stores, use percentage difference, which divides by the average of the two:
percentage difference = |A − B| ÷ ((A + B) ÷ 2) × 100
For 30 and 50: |30 − 50| = 20, the average is 40, and 20 ÷ 40 × 100 = 50%. Compare that with percent change, which depends on direction: 30 → 50 is about +66.67%, and 50 → 30 is −40%.
Special cases
The original value is zero. Percentage change is undefined, since you can't divide by zero. Describe the actual change instead.
The original value is negative. Dividing by a negative base flips the sign and gives a misleading answer. A common convention is to divide by the absolute value of the original: a loss shrinking from −40 to −10 is (−10 − (−40)) ÷ |−40| × 100 = 30 ÷ 40 × 100 = +75%, an improvement. Conventions vary between fields, so say which one you used.
Very small bases. A change from 2 to 6 is a 200% increase, which sounds dramatic but is only 4 units. Report the raw numbers alongside the percentage when the base is small.
Percentage change in a spreadsheet
With the old value in A2 and the new value in B2, enter =(B2-A2)/A2 and format the cell as a percentage. The spreadsheet multiplies by 100 for display, so don't add *100 as well, or you'll get 1250% instead of 12.5%. For data that may include negative starting values, use =(B2-A2)/ABS(A2).
Check your answer
A quick sanity check catches most errors:
- Is the sign right? If the value went down, the answer should be negative (or labeled a decrease).
- Does the size make sense? Doubling should give 100%, halving −50%.
- Apply the percentage to the old value. 64 × 1.125 = 72, so a 12.5% increase from 64 does give 72 ✓.
A percent change calculator shows the difference, the signed percentage and each step, which is handy for checking homework or a long list of values.
Percent Change CalculatorIncrease or decrease between two valuesPercentage change compares two points in time. If you go on to calculus, you'll meet the idea of an instantaneous rate of change, which is what derivative rules let you calculate.
Key takeaways
- Percentage change = (new − old) ÷ old × 100. Positive means increase, negative means decrease.
- Always divide by the starting value, and round only at the end.
- An increase and the decrease that reverses it have different percentages; combine successive changes by multiplying factors.
- Percentage points and percent change are different measures. Say which one you mean.
- To find an original value after a change, divide by the multiplier rather than adding the percentage back.
