Percent
Percent 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," per hundred, so if $30$ out of every $100$ students like math, that is $30$ percent. Formally, a percent is a ratio expressed as a fraction of $100$, $p\% = p/100$, and 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). As a dimensionless ratio scaled by $100$ ($p\% = p \times 10^{-2}$), percent underlies percent error ($|\text{approximate} - \text{exact}|/|\text{exact}| \times 100$) and percent change ($(\text{new} - \text{old})/\text{old} \times 100$) throughout statistics, science, and finance.
Example
$50\% = 50/100 = 1/2$ and $25\% = 25/100 = 1/4$; a score of $80\%$ on a test means $80$ out of $100$ questions correct. A shirt costing $\$40$ discounted $15\%$: discount $= 15/100 \times \$40 = \$6$, so the sale price is $\$34$, the same as computing $85\%$ of $\$40$. The percent error in approximating $\pi$ with $22/7$ is $|22/7 - \pi|/\pi \times 100 \approx 0.04\%$, the same formula used in chemistry to compute percent yield, $(\text{actual yield}/\text{theoretical yield}) \times 100$.
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, the same idea in a convenient costume. Percents are always relative to a base (the "whole"): $50\%$ of $10$ and $50\%$ of $1000$ are very different amounts, so always identify the base before computing, a common source of confusion in real-world problems. 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 same idea generalizes to "parts per million" (ppm) and "parts per billion" (ppb) for very small concentrations, scaled by $10^6$ or $10^9$ instead.