# What is the percent change?

Finds the percentage change from an old value to a new one, or either value from the other and the change.

- Page: https://www.acalculator.org/math/percentage-change-calculator
- JSON spec: https://www.acalculator.org/math/percentage-change-calculator.json
- Version: a3576b0038c5

## Default answer

Example with the default inputs (From (old value) 100, To (new value) 125): From 100 to 125 is a change of 25%.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| from | From (old value) | The starting value A, before the change. Not 0. |
| to | To (new value) | The value B after the change. |
| change | Percentage change | The change from A to B as a percent of \|A\|: positive for an increase, negative for a decrease. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| from | From (old value) | The starting value A, before the change. Not 0. |
| to | To (new value) | The value B after the change. |
| change | Percentage change | The change from A to B as a percent of \|A\|: positive for an increase, negative for a decrease. |
| difference | Difference | The new value minus the old value: B − A. |
| direction | Increase or decrease | Whether the value went up, went down, or stayed the same. |

## Method

p = (B − A) ÷ |A| × 100, where A is the old value, B the new value, and p the percentage change.

## Assumptions

- The change is measured against the size of the old value, |A|, so a rise from −50 to 25 is +150%.
- There is no percentage change from 0: the old value must not be 0.
- Finding the old value from the new value and the change can give two answers, one positive and one negative: both are shown.

## Worked examples

1. from = 100, to = 125 gives change = 25%, difference = 25, direction = Increase. Source: hand calculation in content.mdx: (125 − 100) ÷ 100 × 100 = 25%.
2. from = 80, to = 60 gives change = -25%, difference = -20, direction = Decrease. Source: hand calculation in content.mdx: (60 − 80) ÷ 80 × 100 = −25%.
3. from = 50, change = 50% gives to = 75. Source: hand calculation in content.mdx: 50 + 50% of 50 = 75.
4. to = 75, change = 50% gives from = 50. Source: hand calculation in content.mdx: 75 ÷ 1.5 = 50.
5. from = -50, to = 25 gives change = 150%, difference = 75, direction = Increase. Source: hand calculation in content.mdx: (25 − (−50)) ÷ |−50| × 100 = 75 ÷ 50 × 100 = 150%.
6. to = 10, change = 200% gives from = 3.333333. Source: hand calculation in content.mdx: 10 ÷ 3 = 3.333…, or 10 ÷ (1 − 2) = −10; Python 3: 10 / 3 = 3.3333333333333335.

## FAQ

### How do I calculate percentage change?

Subtract the old value from the new value, divide by the old value, and multiply by 100: (new − old) ÷ old × 100. From 100 to 125 the change is (125 − 100) ÷ 100 × 100 = 25%. A positive answer is an increase and a negative answer is a decrease.

### Is a percentage change the same as a percentage difference?

No. A percentage change has a direction: it measures the new value against the old one, so 100 to 125 is +25% but 125 to 100 is −20%. A percentage difference compares two values with no before and after, against their average, so both orders give 22.22%.

### Why does a 50% rise and a 50% fall not get me back where I started?

Because each change is measured against a different starting value. 100 up 50% is 150, and 150 down 50% is 75. To get from 150 back to 100 you need a fall of 33.33%.

### How do I find the original value before a percentage change?

Divide the new value by 1 plus the change as a decimal. If a price is 75 after a 50% rise, the original was 75 ÷ 1.5 = 50. After a 20% fall to 80, it was 80 ÷ 0.8 = 100.

### How is the percentage change worked out when the old value is negative?

The change is divided by the size of the old value, |old|, so that a rise always shows as a positive percentage. From −50 to 25 is a rise of 75, and 75 ÷ 50 × 100 = +150%. Dividing by −50 instead would give −150%, which would call a rise a decrease.

### What is the percentage change from 0?

There is none. Any change from 0 is infinitely many times 0, so the formula divides by zero. Report the change as an amount (for example, from 0 to 25 is a rise of 25) instead.

### What is the difference between percent change and percentage points?

Percentage points are the plain difference between two percentages. A rate that goes from 4% to 5% rises by 1 percentage point, which is a percentage change of (5 − 4) ÷ 4 × 100 = 25%.

## Sources

- OpenStax, Elementary Algebra 2e, section 3.2 Solve Percent Applications (percent increase and percent decrease): https://openstax.org/books/elementary-algebra-2e/pages/3-2-solve-percent-applications