Percent change

The denominator is the original

This is the only thing in the formula that goes wrong, and it goes wrong often.

Going from $40$ to $50$ is a $25\%$ increase, because the rise of $10$ is compared with the $40$ you started from. Compared with $50$ it would be $20\%$ — the right arithmetic on the wrong base.

A percent change describes how big the change was relative to where it began, so the beginning is the whole.

Why the same two numbers give two different percents

$40 \to 50$ and $50 \to 40$ both move by $10$, but they are different fractions of different starting points:

$$\frac{10}{40} = 25\% \qquad \frac{10}{50} = 20\%$$

So a $25\%$ rise is undone by a $20\%$ fall, not another $25\%$ one. This is the same reason a rise and an equal fall never cancel — each percent is measured against a base that has moved.

Applying a change, rather than measuring one

To apply a percent change, multiply by one plus or minus the decimal.

$$\text{new} = \text{original} \times (1 + r)$$

Change	Multiplier
up $20\%$	$1.2$
up $8\%$	$1.08$
down $20\%$	$0.8$
down $35\%$	$0.65$

Two changes in a row multiply, so a $10\%$ rise then a $20\%$ rise is $1.1 \times 1.2 = 1.32$, a $32\%$ increase — not $30\%$.

Worked examples

A rent rises from $800 to $920.

$$\frac{920 - 800}{800} = \frac{120}{800} = 0.15 \quad\Rightarrow\quad 15\% \text{ increase}$$

A price is $75 after a 25% rise. What was it before?

The new price is $125\%$ of the old, so divide rather than multiply:

$$\frac{75}{1.25} = \$60$$

Taking $25\%$ off $75 gives $56.25, which is a different — and wrong — answer.

Percent change or percentage points?

Subtracting two percents gives percentage points; dividing by the original gives a percent change. A rate moving from $4\%$ to $6\%$ is a rise of $2$ points, or $50\%$. Both are correct answers to different questions.