Percent change calculator
Percent change equals (new value − old value) ÷ old value × 100. Going from 80 to 100 is a 25% increase. The calculator also works in reverse: apply a percentage to a number, recover the original value, and combine several changes in a row using multipliers.
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📌 Key facts
- Percent change equals (new value - old value) / old value x 100.
- The multiplier is 1 + rate: a 20% increase multiplies by 1.2 and a 20% decrease multiplies by 0.8.
- Two successive changes combine by multiplying their multipliers, not by adding the percentages.
- A 20% increase followed by a 20% decrease gives an overall 4% decrease.
- To recover the original value, divide the final value by the multiplier.
The percent change formula
Percent change compares a new value with an old one. It is positive for an increase and negative for a decrease.
\[ \text{percent change} = \dfrac{\text{new value} - \text{old value}}{\text{old value}} \times 100 \]
| Old | New | Work | Change |
|---|---|---|---|
| 80 | 100 | \( \dfrac{20}{80} \times 100 \) | +25% |
| 100 | 80 | \( \dfrac{- 20}{100} \times 100 \) | −20% |
| 50 | 150 | \( \dfrac{100}{50} \times 100 \) | +200% |
Note that going from 80 to 100 is up 25%, but going from 100 to 80 is down 20%. The percent always depends on the starting value.
Multipliers and successive percent changes
Increasing by r% means multiplying by \( 1 + \dfrac{r}{100} \); decreasing by r% means multiplying by \( 1 - \dfrac{r}{100} \). That number is the multiplier.
| Change | Multiplier |
|---|---|
| +5% | 1.05 |
| +50% | 1.5 |
| −20% | 0.8 |
| −100% | 0 |
For two changes in a row, multiply the multipliers. A 10% increase followed by a 10% decrease gives \( 1.1 \times 0.9 = 0.99 \), a 1% decrease overall, not a return to the start. The reverse of a 25% increase is a 20% decrease \( \left( \dfrac{1}{1.25} = 0.8 \right) \).
Finding the original value
If a price is $96 after a 20% discount, the original is not $96 plus 20% of $96. Divide by the multiplier instead.
\[ \text{original value} = \dfrac{\text{final value}}{1 + \dfrac{\text{percent}}{100}} \]
Here \( \dfrac{96}{0.8} = \$120 \). Check: \( 120 \times 0.8 = 96 \). The same method finds a pre-tax price from a total that already includes sales tax.
Percent change: reading the sign and the size
The sign of the percent change shows the direction. A positive value means an increase, and a negative value means a decrease.
| Old value | New value | Percent change |
|---|---|---|
| 200 | 230 | +15% |
| 80 | 60 | -25% |
| 50 | 100 | +100% |
So doubling a value is a 100% increase, while cutting it in half is a 50% decrease.
Percent change: successive changes
When two changes happen in a row, multiply the multipliers. For example, +50% then -50% gives 1.5 x 0.5 = 0.75, an overall 25% decrease.
Therefore, a rise followed by a fall of the same percentage never brings you back to the start. However, +25% then -20% does return to the start, because 1.25 x 0.8 = 1.
Then read the overall percent change from the final multiplier: 0.75 means -25%.
Percent change versus percentage points
Do not mix up percent and percentage points. If an unemployment rate moves from 8% to 10%, it gains 2 percentage points. Its percent change, though, is +25%, because 2 / 8 = 0.25.
Also, a 30% discount means you multiply by 0.7. After all, 100 - 30 = 70, which is 0.7. To find the original price of an item that costs $70 after a 30% discount, divide by 0.7: $100.
Percent change: more tools
To average class grades with weights, use the weighted average calculator. For a fraction, open the fraction calculator. Finally, for powers and roots, try the scientific calculator or the equation solver.
Percent change: real-life uses
Percent change shows up every day. For example, a price moves from $40 to $46: 6 / 40 = 0.15, which is +15%. So you quickly grasp the effect of a rise.
Then compare series. If a town's population drops from 12,000 to 11,400, the rate is -600 / 12,000 = -5%. Also, an average yearly rate uses a root: over three years, take the cube root of the total multiplier. Finally, always sanity-check the size of the result. A 150% raise would be surprising, so reread your entries.
In school, these problems appear often in percent chapters.
Percent change: checking your result
To verify your work, run it backward. First, apply the multiplier to the starting value. Next, compare with the ending value. Finally, fix any mismatch.
For example, from 250 to 300 the multiplier is 1.2. So 250 x 1.2 = 300, and the result is right. Also, check the sign. A value that falls must give a negative rate. A common mistake, however, is the choice of base. The rate always uses the starting value, never the final one. So going from 100 to 50 is -50%, while going from 50 to 100 is +100%.
3 real-life examples
Rent that rises from $1,250 to $1,325
\( \dfrac{1{,}325 - 1{,}250}{1{,}250} \times 100 = \dfrac{75}{1{,}250} \times 100 = \) +6%. The multiplier is \( \dfrac{1{,}325}{1{,}250} = 1.06 \).
A 30% off sale on a $250 jacket
The multiplier is \( 1 - 0.30 = 0.7 \). Sale price: \( 250 \times 0.7 = \) $175, which saves $75.
A 10% increase then a 10% decrease
\( 1.1 \times 0.9 = 0.99 \). Overall that is a 1% decrease: a $100 item ends at $99, not $100.