Percentage
Fractions & DecimalsA 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 $\%$, you are looking at a percentage. It tells you how much of something you have out of $100$ equal parts.
Example
If $15$ out of $50$ students have a pet, what percentage is that? $15/50 = 30/100 = 30\%$. Thirty percent of the students have a pet.
Key Insight
The difference between "percent" and "percentage" is subtle: percent refers to the symbol or rate (like "$30$ percent"), while percentage refers to the result or quantity (like "the percentage of students is $30\%$"). In practice, they are often used interchangeably.
Definition
A percentage expresses a ratio as a fraction of $100$. The three key quantities in percentage problems are: Part ($P$), Whole ($W$), and Percent Rate ($R\%$). They relate by: $P = R/100 \times W$. Any one quantity can be found if the other two are known: $R = P/W \times 100$; $W = P/(R/100)$.
Example
What is $35\%$ of $80$? $P = 35/100 \times 80 = 28$. A score of $54$ out of $60$: $R = 54/60 \times 100 = 90\%$. If $40\%$ of a number is $26$, what is the number? $W = 26/0.40 = 65$.
Key Insight
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. Drawing a proportion $P/W = R/100$ makes it visual and mechanical.
Definition
Percentage is a linear transformation of a ratio: $f(r) = 100r$, mapping $r$ in $[0,1]$ to $[0\%,100\%]$ (or beyond for rates $> 1$). In statistics, percentages appear as relative frequencies; in calculus, as local linear approximations ($df/f \times 100$ gives percent change). Percentage points differ from percent change: a rise from $10\%$ to $15\%$ is $5$ percentage points but a $50\%$ relative increase.
Example
If an investment grows from $\$1000$ to $\$1200$, the percentage increase is $(200/1000) \times 100 = 20\%$. If the interest rate rises from $3\%$ to $4\%$, that is a $1$ percentage point increase but a $33.3\%$ relative increase in the rate itself. Confusing the two is common in financial reporting.
Key Insight
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.