Percentage Increase Formula - Definition, Steps, Examples

Key Points

• The percentage increase formula is $\frac{\text{new} - \text{old}}{\text{old}} \times 100$, dividing the rise by the original value.

• The denominator is always the original value, because an increase is measured from where the quantity started.

• Rearranged, the new value after a rise of $p$ per cent is $\text{new} = \text{old}\left(1 + \frac{p}{100}\right)$.

• Successive increases multiply rather than add, so two rises of 10 per cent give 21 per cent overall, not 20.

• A percentage increase has no upper limit, unlike a percentage decrease, which cannot exceed 100 per cent.

What Is the Percentage Increase Formula?

The percentage increase formula expresses a rise as a proportion of what came before:

$$\text{percentage increase} = \frac{\text{new} - \text{old}}{\text{old}} \times 100$$

The numerator is the rise and the denominator is the starting value. The US Bureau of Labor Statistics applies this logic to index numbers, reporting that percent change in the index estimates the percent change in prices, with 1-month and 12-month changes commonly published.

Why the Original Value Is the Denominator

An increase is a journey from one value to another, and the percentage describes how big that journey is compared with the starting point. Dividing by the destination would answer a different question.

The asymmetry is easy to see. Going from 40 to 50 is a rise of 10 on a base of 40, which is 25 per cent, while going from 50 to 40 is a fall of 10 on a base of 50, which is 20 per cent.

Same ten units, two different percentages, because the bases differ. That is why a rise and the matching fall never cancel.

Variable Key

Examples of the Percentage Increase Formula

Example 1

A price rises from 40 to 50. Find the percentage increase.

$$\frac{50 - 40}{40} \times 100 = \frac{10}{40} \times 100 = 25$$

Final answer: 25 per cent

Example 2

A salary rises from 40,000 to 50,000. Find the percentage increase by dividing by the new value.

The new figure is the one being reported, so it looks like the natural reference:

$$\frac{10000}{50000} \times 100 = 20$$

That answers a different question. Dividing by the new value tells you what share of the final salary the rise represents, not how much the original salary grew.

An increase is measured from where it started, so the denominator is the old value:

$$\frac{10000}{40000} \times 100 = 25$$

Final answer: 25 per cent

Example 3

A population of 1,250 grows by 8 per cent. Find the new population.

$$1250\left(1 + \frac{8}{100}\right) = 1250 \times 1.08 = 1350$$

Final answer: 1,350

Example 4

A value rises 10 per cent, then rises 10 per cent again. Find the overall percentage increase.

The two rises act on different bases, so multiply the growth factors:

$$1.10 \times 1.10 = 1.21$$

Final answer: 21 per cent, not 20. The second rise is taken on the already-increased value.

Example 5

A quantity increased by 35 per cent to reach 540. Find the original value.

The final value is 135 per cent of the original, so divide:

$$\frac{540}{1.35} = 400$$

Final answer: 400

Subtracting 35 per cent of 540 would give 351, which is wrong because that takes the percentage on the new value.

Don't Confuse These Four Change Measures

The percentage point is the one that causes most trouble in reporting. A rate moving from 4 per cent to 5 per cent is a rise of one percentage point but a percentage increase of 25 per cent, and the two are often used interchangeably in error.

Where Intuition Breaks on Percentage Increase

Mistake 1: Dividing by the new value

Where it slips in: The new value is usually the headline figure, so it becomes the reference by default.

Don't do this: $\frac{\text{rise}}{\text{new}} \times 100$

The correct way: $\frac{\text{rise}}{\text{old}} \times 100$. The base is where the quantity started, which is also why the result can exceed 100 per cent.

Students who write "from" and "to" above the two numbers before computing anything pick the right denominator automatically, because the base is whatever follows "from". It is the same habit that fixes percentage decrease.

Mistake 2: Adding successive increases

Where it slips in: Two rises in sequence look additive.

Don't do this: Treat 10 per cent then 10 per cent as 20 per cent.

The correct way: Multiply the factors, $1.10 \times 1.10 = 1.21$, giving 21 per cent. The second rise is applied to a larger quantity than the first.

Mistake 3: Confusing a percentage increase with a percentage point change

Where it slips in: The original quantity is itself a percentage, so both readings sound valid.

Don't do this: Describe a move from 4 per cent to 5 per cent as a 1 per cent increase.

The correct way: Call it one percentage point, or a 25 per cent increase, and say which. Mixing them understates a change by a factor of 25 in this example.

Bottom Line

• The percentage increase formula is $\frac{\text{new} - \text{old}}{\text{old}} \times 100$.

• The original value is always the denominator, which is what makes a rise and the matching fall differ.

• Successive increases multiply their growth factors, so two 10 per cent rises total 21 per cent.

• A percentage increase and a percentage point change are different statements about the same movement.

• To work through percentages with a teacher, explore a 6th grade math tutor or math classes online.

Where to Go From Here

Find the percentage increase from 80 to 92. Then apply three successive 5 per cent rises to 1,000 and compare the result against a single 15 per cent rise.

Want a live Bhanzu trainer to walk through more percentage problems? Book a free demo class.

Read More

• Percent Decrease Formula

• Percent Difference Formula

• Discount Formula

• What Is a Percent

• How to Teach Percentages