Percentage change answers one question: how big is the difference between two numbers, measured against the number you started with? The arithmetic is short, but the details trip people up. Which number goes on the bottom, why a 50% rise followed by a 50% fall leaves you worse off, and why "up 1%" and "up 1 percentage point" mean different things. This guide covers the formulas, then works through real examples you can check line by line.
The one formula behind all of it
Every percentage increase, decrease and change uses the same formula:
percentage change = (new value − old value) ÷ old value × 100- If the result is positive, it is a percentage increase.
- If the result is negative, it is a percentage decrease (usually reported without the minus sign, as "a 20% decrease").
The key word is old. You always divide by the starting value, the baseline you are measuring from. Dividing by the new value is the most common mistake in percentage questions, and it gives a different answer.
Percentage increase: worked examples
A price rise
A box of printer paper went from $40 to $52.
- Difference: 52 − 40 = 12
- Divide by the old value: 12 ÷ 40 = 0.30
- Multiply by 100: 0.30 × 100 = 30% increase
Check: 40 × 1.30 = 52. Multiplying the old value by (1 + the rate) should always give you the new value back. It is a good habit for catching slips.
A salary raise
Your salary goes from $58,000 to $61,190.
- Difference: 61,190 − 58,000 = 3,190
- 3,190 ÷ 58,000 = 0.055
- 0.055 × 100 = 5.5% raise
Check: 58,000 × 1.055 = 61,190.
Adding a percentage to a number
Sometimes you know the rate and want the result. To increase 250 by 8%, multiply by 1.08:
250 × 1.08 = 270That is the same as working out 8% of 250 (which is 20) and adding it on, but one multiplication is quicker and easier to put in a spreadsheet. The percentage increase calculator does both directions and shows each step.
Percentage decrease: worked examples
A discount
A jacket is reduced from $80 to $60.
- Difference: 60 − 80 = −20
- −20 ÷ 80 = −0.25
- 25% decrease, or "25% off"
To take a percentage off directly, multiply by (1 − the rate). For 25% off $80: 80 × 0.75 = 60.
A population decline
A region's population falls from 2.40 million to 2.31 million.
- Difference: 2.31 − 2.40 = −0.09 million
- −0.09 ÷ 2.40 = −0.0375
- 3.75% decrease
Notice you can work in millions throughout. As long as both numbers use the same unit, the units cancel out. The percentage decrease calculator handles discounts and drops like these.
Direction matters: the same gap gives two answers
A student's test score rises from 64 to 80:
(80 − 64) ÷ 64 = 16 ÷ 64 = 0.25 → 25% increaseNow run it backwards. If the score falls from 80 to 64:
(64 − 80) ÷ 80 = −16 ÷ 80 = −0.20 → 20% decreaseThe gap is 16 points both times, but the baseline changes, so the percentage changes. This is why "the price went up 25%" and "the price was 20% lower before" can describe the same two numbers. When you report a change, be clear which value is the starting point. A percentage change calculator lets you enter the old and new values and gives the signed result, so the direction is never ambiguous.
The trap: +50% then −50% does not get you back
Suppose an investment of $100 rises 50%, then falls 50%.
- After the rise: 100 × 1.50 = 150
- After the fall: 150 × 0.50 = 75
You end at $75, a 25% loss overall, not back at $100. The fall is 50% of a bigger number than the rise was 50% of.
The general rule: successive percentage changes multiply, they don't add. Convert each change to a multiplier and multiply them together:
| Changes | Multipliers | Combined | Net effect |
|---|---|---|---|
| +50%, then −50% | 1.50 × 0.50 | 0.75 | −25% |
| +10%, then −10% | 1.10 × 0.90 | 0.99 | −1% |
| +20%, then +20% | 1.20 × 1.20 | 1.44 | +44% (not +40%) |
| −20%, then +25% | 0.80 × 1.25 | 1.00 | 0% (back to start) |
The last row shows the flip side: after a loss, you need a larger percentage gain to recover. A 20% drop needs a 25% rise to break even, and a 50% drop needs a 100% rise. The recovery needed is:
recovery = 1 ÷ (1 − loss) − 1For a 20% loss: 1 ÷ 0.80 − 1 = 1.25 − 1 = 0.25, or 25%.
Reverse percentages: finding the original value
A common question is "the price is $69 after a 15% increase. What was it before?"
The tempting answer is to take 15% off $69:
69 × 0.85 = 58.65 ← wrongThat is wrong because the 15% was calculated on the original price, not on $69. The new price is the original × 1.15, so you divide to undo it:
original = 69 ÷ 1.15 = 60Check: 60 × 1.15 = 69. The original price was $60.
The same logic works for discounts. If you paid $36 after a 20% discount, the original price was:
36 ÷ 0.80 = 45Check: 45 × 0.80 = 36.
The rule: to reverse a percentage change, divide by the multiplier you would have used to apply it. This comes up constantly with sales tax and VAT, where you have a tax-inclusive price and need the pre-tax amount. With a 20% tax rate, a $120 price is $120 ÷ 1.20 = $100 before tax, not $120 × 0.80 = $96.
Percentage points vs percent
When the numbers you are comparing are already percentages, such as interest rates, unemployment rates or exam pass rates, there are two ways to describe a change, and they are not interchangeable.
- Percentage points are the simple difference between two percentages.
- Percent is the relative change, using the formula above.
Example: a savings rate goes from 4% to 5%.
- Change in percentage points: 5 − 4 = 1 percentage point
- Percentage change: (5 − 4) ÷ 4 = 0.25 = 25% increase
Another: unemployment falls from 5.0% to 4.5%.
- That is a fall of 0.5 percentage points
- As a percentage change: (4.5 − 5.0) ÷ 5.0 = −0.10, a 10% decrease
Both statements are true, but "unemployment fell 10%" sounds far more dramatic than "unemployment fell half a point". When you read a headline about a rate, check which one is being used. When you write one, use "percentage points" (sometimes abbreviated "pp") for the simple difference.
Edge cases worth knowing
The old value is zero. You can't divide by zero, so percentage change from zero is undefined. Going from 0 sales to 15 sales is not an "infinite" increase in any useful sense. Report the absolute change instead ("up from 0 to 15").
The old value is negative. If a company's profit goes from −$200 to $100, the formula gives (100 − (−200)) ÷ −200 = −150%, which suggests a decline when the result clearly improved. A common workaround is to divide by the absolute value of the old number, which gives +150%, but many analysts prefer to avoid percentages entirely when the baseline is negative and just state the change in dollars.
Comparing two values with no "before". If you are comparing two things side by side, like two shops' prices, rather than one thing over time, there is no natural baseline. Some fields use percentage difference instead, which divides by the average of the two: for $40 and $52, that is 12 ÷ 46 ≈ 26.09%. Say which method you used.
Spreadsheet formulas
These work in Excel, Google Sheets and LibreOffice Calc. Assume the old value is in A2 and the new value is in B2. Format the result cell as a percentage so 0.3 displays as 30%.
Percentage change: =(B2-A2)/A2
Same, guarding against zero: =IF(A2=0, "n/a", (B2-A2)/A2)
Negative baselines: =(B2-A2)/ABS(A2)To apply or reverse a rate, put the rate in C2 (as 15% or 0.15):
Increase A2 by C2: =A2*(1+C2)
Decrease A2 by C2: =A2*(1-C2)
Original before an increase: =B2/(1+C2)
Original before a discount: =B2/(1-C2)A frequent spreadsheet mistake is typing =A2*15%+A2 in one place and =A2*1.15 in another. Both are correct, but mixing styles makes errors harder to spot. Pick the multiplier form and use it everywhere.
Quick reference
| You want to… | Formula | Example |
|---|---|---|
| Find the % change | (new − old) ÷ old × 100 | 40 → 52 is +30% |
| Increase by a % | value × (1 + rate) | 250 + 8% = 270 |
| Decrease by a % | value × (1 − rate) | 80 − 25% = 60 |
| Undo an increase | value ÷ (1 + rate) | 69 before +15% = 60 |
| Undo a decrease | value ÷ (1 − rate) | 36 before −20% = 45 |
| Combine changes | multiply the multipliers | +50% then −50% = −25% |
Check your numbers in seconds
For a one-off answer with the working shown, use the percentage change calculator: enter the old and new values and it tells you the size and direction of the change. If you are adding or removing a known percentage, the increase and decrease calculators handle that directly. All three run in your browser, are free, and need no account.
FAQ
How do I calculate a percentage increase between two numbers?
Subtract the old number from the new one, divide by the old number, and multiply by 100. From 40 to 52: (52 − 40) ÷ 40 × 100 = 30%.
Is percentage decrease calculated the same way?
Yes. It is the same formula, and the result comes out negative. From 80 to 60: (60 − 80) ÷ 80 × 100 = −25%, which you report as a 25% decrease.
Why doesn't a 50% increase followed by a 50% decrease cancel out?
Because the decrease is applied to the larger, already-increased number. 100 becomes 150, and 50% of 150 is 75, so you end at 75. Percentage changes combine by multiplying (1.5 × 0.5 = 0.75), not by adding.
How do I find the original price before a percentage increase?
Divide the final price by (1 + the rate). A $69 price after a 15% increase was $69 ÷ 1.15 = $60. Taking 15% off $69 gives $58.65, which is wrong.
What is the difference between percent and percentage points?
Percentage points are the simple difference between two percentages: 4% to 5% is up 1 percentage point. Percent is the relative change: 4% to 5% is a 25% increase, because 1 is a quarter of 4.