# What is the percentage decrease?

Finds the percentage decrease from an original value to a new value, P = (V₁ − V₂) ÷ V₁ × 100, or the new or original value from the percent.

- Page: https://www.acalculator.org/math/percentage-decrease-calculator
- JSON spec: https://www.acalculator.org/math/percentage-decrease-calculator.json
- Version: c6cdb3fa3a13

## Default answer

Example with the default inputs (Original value 50, New value 40): From 50 to 40 is a 20% decrease.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| from | Original value | The value before the decrease. It must be greater than 0. |
| to | New value | The value after the decrease. It cannot be negative. |
| percent | Percentage decrease | How much smaller the new value is, as a percent of the original value. A negative percent is an increase. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| from | Original value | The value before the decrease. It must be greater than 0. |
| to | New value | The value after the decrease. It cannot be negative. |
| percent | Percentage decrease | How much smaller the new value is, as a percent of the original value. A negative percent is an increase. |
| change | Amount of decrease | The original value minus the new value: V₁ − V₂. Negative when the value went up. |
| verdict | In words | The change in words: a decrease, or an increase when the new value is larger than the original. |

## Method

P = (V₁ − V₂) ÷ V₁ × 100, where V₁ is the original value and V₂ the new value.

## Assumptions

- The percent is taken of the original value, not of the new value.
- The original value is greater than 0 and the new value is 0 or more, so the decrease is at most 100%.
- A new value larger than the original gives a negative decrease, and the answer calls it an increase.
- The original value cannot be found from a 100% decrease, because every original value falls to 0.

## Worked examples

1. from = 50, to = 40 gives percent = 20%, change = 10, verdict = a 20% decrease. Source: hand calculation in content.mdx: (50 − 40) ÷ 50 × 100 = 20%.
2. from = 3.71, to = 3.64 gives percent = 1.886792%. Source: OpenStax Prealgebra 2e, Example 6.24: a gas price from $3.71 to $3.64 is a 1.9% decrease (rounded there); (3.71 − 3.64) ÷ 3.71 × 100 = 1.8868%.
3. from = 80, percent = 25% gives to = 60, change = 20. Source: hand calculation in content.mdx: 80 × (1 − 0.25) = 60.
4. to = 60, percent = 25% gives from = 80. Source: hand calculation in content.mdx: 60 ÷ (1 − 0.25) = 80.
5. from = 40, to = 50 gives percent = -25%, change = -10, verdict = a 25% increase, not a decrease. Source: hand calculation in content.mdx: (40 − 50) ÷ 40 × 100 = −25%, an increase of 25%.

## FAQ

### How do I calculate a percentage decrease?

Subtract the new value from the original, divide by the original, and multiply by 100: (original − new) ÷ original × 100. A price that falls from $50 to $40 has dropped (50 − 40) ÷ 50 × 100 = 20%.

### How do I take a percentage off a number?

Multiply by 1 minus the percent as a decimal. 25% off $80 is 80 × (1 − 0.25) = 80 × 0.75 = $60.

### How do I find the original price before a discount?

Divide the sale price by 1 minus the percent as a decimal. A $60 price after 25% off was 60 ÷ 0.75 = $80. Adding 25% back to $60 gives $75, which is wrong, because the 25% was taken from $80, not from $60.

### Why is a 20% fall not undone by a 20% rise?

The rise is taken from the smaller number. 50 down 20% is 40, and 40 up 20% is only 48. To get from 40 back to 50 you need a 25% rise.

### What does a negative percentage decrease mean?

The value went up. From 40 to 50 is a decrease of −25%, which is an increase of 25% of the original 40.

### Can a decrease be more than 100%?

Not for amounts that cannot go below zero, such as prices, weights, or counts. A 100% decrease takes the value to 0, so the calculator accepts decreases of up to 100%.

### What is the difference between a percentage decrease and a percentage point decrease?

A rate that falls from 5% to 4% has dropped 1 percentage point, which is a 20% decrease of the rate: (5 − 4) ÷ 5 × 100 = 20%.

## Sources

- OpenStax, Prealgebra 2e, §6.2 Solve General Applications of Percent (percent decrease: decrease = original amount − new amount, as a percent of the original; Example 6.24). https://openstax.org/books/prealgebra-2e/pages/6-2-solve-general-applications-of-percent