What Is a Percent? Percents, Fractions and Decimals

Per hundred

Percent means per hundred. The word comes from the Latin per centum, and the sign itself is a squashed-up $100$ — the $1$ flattened into a slash, with the two zeros left either side.

So $30\%$ means 30 out of every 100.

That is the whole definition. Everything else in this lesson, and in every percent lesson after it, is a consequence of it.

Why a hundred, and not something else

Fractions can be built on any denominator, and that flexibility is exactly what makes them awkward to compare. Is $\tfrac{17}{40}$ bigger than $\tfrac{5}{12}$? Nothing about those numbers tells you at a glance.

Percent fixes the denominator at $100$ for everybody. Once two quantities are both “out of 100”, comparing them is counting.

Written as a fraction	As a percent
$\tfrac{17}{40}$	$42.5\%$
$\tfrac{5}{12}$	$41.\overline{6}\%$

The percents answer the question in one glance. That is the entire reason the notation exists: it is a shared denominator for the whole world.

The same number, three ways

$$30\% \;=\; \frac{30}{100} \;=\; 0.3$$

These are not three facts to memorise. They are one number in three costumes, and the conversions between them follow from what percent means.

To go from	To	Do this	Example
percent	fraction	write it over $100$, then simplify	$30\% = \tfrac{30}{100} = \tfrac{3}{10}$
percent	decimal	divide by $100$	$30\% = 0.3$
decimal	percent	multiply by $100$	$0.3 = 30\%$
fraction	percent	convert to a decimal first, then multiply by $100$	$\tfrac{3}{4} = 0.75 = 75\%$

Why the decimal point moves two places

Dividing by $100$ moves the point two places left; multiplying by $100$ moves it two places right. This is not a rule about percents at all — it is how place value works.

Each place in a decimal is worth ten times the one to its right. Dividing by $10$ shifts every digit one place down; dividing by $100$ is doing that twice.

$$45\% \;\rightarrow\; 0.45 \qquad 7\% \;\rightarrow\; 0.07 \qquad 250\% \;\rightarrow\; 2.5$$

The zero in $0.07$ is doing real work. Without it you get $0.7$, which is ten times too big.

The ones worth knowing on sight

These come up constantly, and recognising them saves real time.

Percent	Fraction	Decimal
$10\%$	$\tfrac{1}{10}$	$0.1$
$20\%$	$\tfrac{1}{5}$	$0.2$
$25\%$	$\tfrac{1}{4}$	$0.25$
$50\%$	$\tfrac{1}{2}$	$0.5$
$75\%$	$\tfrac{3}{4}$	$0.75$
$100\%$	$1$	$1$
$12.5\%$	$\tfrac{1}{8}$	$0.125$
$33.\overline{3}\%$	$\tfrac{1}{3}$	$0.\overline{3}$

Notice that $100\%$ is $1$ — the whole thing. That single fact explains why $50\%$ is a half and why anything above $100\%$ is more than you started with.

Worked examples

Common mistakes

Practice problems