Percent
Fractions & DecimalsPercent means "per hundred" and is a way to express a ratio or fraction as a number out of 100, written with the % symbol.
Formula
\text{percent} = \left(\frac{\text{part}}{\text{whole}}\right) \times 100
Definition
Percent means "out of $100$." The $\%$ symbol is short for "per cent," which means "per hundred." If $30$ out of every $100$ students like math, that is $30$ percent ($30\%$).
Example
$50\% = 50/100 = 1/2$. $25\% = 25/100 = 1/4$. $100\%$ means the whole thing. $0\%$ means none of it. A score of $80\%$ on a test means $80$ out of $100$ questions correct.
Key Insight
Percent is just a fraction with a denominator of $100$, written in a special way. Everything you know about fractions applies to percents - they are the same idea in a convenient costume.
Definition
A percent is a ratio expressed as a fraction of $100$: $p\% = p/100$. Converting between forms: $p\% = p/100$ (fraction) $= p/100$ as a decimal. The formula connecting part, whole, and percent is: $\text{part} = \text{percent}/100 \times \text{whole}$. Percents can exceed $100\%$ (more than the whole) or be less than $1\%$ (very small portions).
Example
A shirt costs $\$40$, discounted $15\%$. Discount $= 15/100 \times \$40 = \$6$. Sale price $= \$40 - \$6 = \$34$. Or: $100\% - 15\% = 85\%$ of $\$40 = 0.85 \times \$40 = \$34$.
Key Insight
Percents are always relative to a base (the "whole"). $50\%$ of $10$ and $50\%$ of $1000$ are very different amounts. Always identify the base before computing a percent - a common source of confusion in real-world problems.
Definition
A percent is a dimensionless ratio scaled by $100$. Formally, $p\% = p \times 10^{-2}$. In statistics, relative frequency and probability are expressed as percents of a total count or sample space. In calculus, percent error $= |\text{approximate} - \text{exact}|/|\text{exact}| \times 100$ quantifies relative error. Percent change $= (\text{new} - \text{old})/\text{old} \times 100$ measures relative growth or decay.
Example
Percent error in the approximation $\pi \approx 22/7$: $|22/7 - \pi|/\pi \times 100 = |3.142857 - 3.141593|/3.141593 \times 100 \approx 0.04\%$. In chemistry, percent yield $= (\text{actual yield}/\text{theoretical yield}) \times 100$.
Key Insight
Percent is the standard unit of relative change in finance, science, and statistics because it removes units and scale, enabling comparison across different quantities. The concept generalizes to "parts per million" (ppm) and "parts per billion" (ppb) for very small concentrations - the same ratio idea scaled by $10^6$ or $10^9$.