Percentage
A percentage is a specific amount or portion expressed as a rate per hundred, often used to describe how much of a whole a part represents.
Formula
\text{percentage} = \left(\frac{\text{part}}{\text{whole}}\right) \times 100\%
Definition
A percentage is an amount stated as part of $100$: when you see a number followed by $\%$, it tells you how much of something you have out of $100$ equal parts. The three key quantities in percentage problems are Part ($P$), Whole ($W$), and Percent Rate ($R\%$), related by $P = R/100 \times W$, so any one quantity can be found once the other two are known. Formally, percentage is a linear transformation of a ratio, $f(r) = 100r$, mapping $r$ in $[0,1]$ to $[0\%,100\%]$ (or beyond for rates greater than $1$); percentage points differ from percent change, a rise from $10\%$ to $15\%$ is $5$ percentage points but a $50\%$ relative increase, a distinction frequently confused in financial and public reporting.
Example
If $15$ out of $50$ students have a pet, the percentage is $15/50 = 30/100 = 30\%$. What is $35\%$ of $80$? $P = 35/100 \times 80 = 28$; and if $40\%$ of a number is $26$, the number is $26/0.40 = 65$. If an investment grows from $\$1000$ to $\$1200$, the percentage increase is $(200/1000) \times 100 = 20\%$, but if an interest rate rises from $3\%$ to $4\%$, that is only a $1$ percentage-point increase even though it is a $33.3\%$ relative increase in the rate itself.
Key Insight
The difference between "percent" and "percentage" is subtle: percent refers to the symbol or rate ("$30$ percent"), while percentage refers to the resulting quantity ("the percentage of students is $30\%$"), though in practice they are often used interchangeably. All percentage problems reduce to the same formula with three variables, recognizing which two you know (and which you are solving for) is the entire skill, and drawing the proportion $P/W = R/100$ makes it visual and mechanical. The distinction between percentage points and percent change is one of the most misused concepts in public discourse: a policy that reduces a disease rate from $8\%$ to $4\%$ achieves a $4$ percentage-point reduction, but a $50\%$ relative reduction, both accurate but giving very different impressions of magnitude.