Percentage Increase / Decrease Formula

A jacket goes from $80$ to $100$. Did it rise $20\%$ or $25\%$? The percentage change formula settles it by comparing the change against the starting value, not the new one.

The formula and its symbols

The numerator is the change, the denominator is the original value, and the sign reads the direction: positive is a percentage increase, negative is a percentage decrease. The formula is defined only when the original value is not $0$, and in most school and everyday examples that value is also positive.

Why the original value sits in the denominator

Percentage change means "change relative to where you started," so the base has to be the starting point. A change of $20$ is large against a start of $80$ but small against a start of $400$ — same change, very different significance. Dividing by the original value scales the change to its starting size, which is exactly the comparison you want. Divide by the new value instead and you are measuring a different ratio that no longer answers "how big was this change compared with where it began."

Worked example: from $80$ to $100$

Start with the change:

Compare it with the original price:

Convert to a percent:

So the price increased by $25\%$. Now reverse it. Going from $100$ down to $80$:

So $100 \to 80$ is a $20\%$ decrease, not $25\%$. The percentages differ because the starting values differ — exactly the point of putting the original value underneath.

Practice with a built-in check

Set up a price dropping from $60$ to $45$. The change is $45 - 60 = -15$; dividing by the original gives $\tfrac{-15}{60} = -0.25$, so it is a $25\%$ decrease. Reading the sign confirms the direction: negative means decrease. As a routine, run four steps every time — find the change, divide by the original, multiply by $100\%$, read the sign.

Calculation traps

Using the wrong denominator. The base must be the original value; dividing by the new value measures a different ratio.

Assuming the reverse percent matches. A $25\%$ increase is not undone by a $25\%$ decrease, because the starting value changes between the two calculations.

Forgetting the zero condition. If the original value is $0$, the formula is undefined, since division by $0$ is not allowed.

You will see percentage increase and decrease in prices, discounts, population changes, test scores, and simple business or science reports — anywhere you want to compare changes fairly across different starting values.