Every formula in this chapter is built on one idea: a rate is a ratio, and a ratio has no units. Percent change is the plainest version — $c = \dfrac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}}$, read aloud as c equals x-new minus x-old, over x-old. Naming the parts: $x_{\text{old}}$ is the value before the change, $x_{\text{new}}$ is the value after it, both in dollars (or litres, or students — the units cancel), and $c$ is the relative change, a bare number. Multiply $c$ by 100 and it becomes a percent.

The denominator is the whole argument. A price that goes from $200 to $240 rose by 20% of where it STARTED; measured against where it ended, the same $40 is only 16.7%. Both fractions are arithmetic; only one of them is percent change. When a headline says “up 20%”, ask immediately: up 20% of what?