Percentages

A percentage is a hundredth of a whole — rather than writing a fraction with a fixed denominator of $100$, we use the $\%$ sign:

For example:

A percentage is simply another way of writing a fraction whose denominator is always $100$.

Percentage of a number

To find how much $p$ percent of a number $c$ is, multiply $c$ by the fraction $\frac{p}{100}$:

For example $20\%$ of $150$ is:

Whole from a known part

Sometimes you only know a part and what percentage of the whole it represents, and you want the whole. Divide the part by the fraction the percentage represents:

If $30$ is $20\%$ of some number, the whole is:

Percentage change

Percentage change tells you how much a value increased or decreased relative to its starting point:

A positive result means an increase, a negative one a decrease. A price rising from $80$ to $100$ changes by $\frac{100 - 80}{80} \cdot 100\% = 25\%$.

Watch out for a common trap: a $20\%$ increase followed by a $20\%$ decrease does not return to the starting point — each percentage is taken from a different (current) base. A $20\%$ increase on $100$ gives $120$; a $20\%$ decrease on $120$ gives $96$, not $100$.

Common mistakes

• Confusing "percentage of a number" with "what percentage one number is of another" — these are two different questions: one multiplies, the other divides.
• Computing percentage change from the new value instead of the old one — the denominator is always the starting value, never the ending one.
• Assuming an increase and a decrease of the same percentage cancel out — they do not, because the base the percentage is taken from changes between the two steps.