Percentages

1. What "per cent" means

The word is Latin: per centum, "out of a hundred." So $7$ percent — written $7\%$ — means "$7$ out of $100$." It's a fraction, just with the denominator pre-decided.

This is why the world reports interest rates, exam scores, sale discounts, polling margins, and most other "fraction-of-a-whole" quantities in percent. They could all be reported as fractions or decimals — they're literally the same numbers — but $13.5\%$ is easier to compare to $17\%$ than $\tfrac{27}{200}$ is to $\tfrac{17}{100}$, even though they're saying the same things.

2. Percent, fraction, and decimal — three names for one idea

The conversion between the three is mechanical. Choose whichever form is most convenient for the operation you're about to do.

The conversion rules in one breath:

• Percent → decimal: divide by $100$ (move the decimal point two places left). $42\% = 0.42$.
• Decimal → percent: multiply by $100$ (move the decimal point two places right). $0.04 = 4\%$.
• Percent → fraction: put it over $100$ and reduce. $35\% = \tfrac{35}{100} = \tfrac{7}{20}$.
• Fraction → percent: divide top by bottom, multiply by $100$. $\tfrac{3}{8} = 0.375 = 37.5\%$.

Percentages can exceed $100$, by the way. $200\%$ is just $2$. "The population grew by $300\%$" means it quadrupled (original plus three more originals). Percentages are not bounded by $100$ — that's a common but incorrect intuition.

3. Percent of a quantity

The single most useful percent operation is "take $P$ percent of $Q$." It just means: multiply $Q$ by $\tfrac{P}{100}$, or equivalently by $P$'s decimal form.

So:

• $25\%$ of $80$ is $\tfrac{25}{100} \cdot 80 = \tfrac{1}{4} \cdot 80 = 20$.
• $15\%$ of $200$ is $0.15 \cdot 200 = 30$.
• $8.25\%$ of $\$120$ is $0.0825 \cdot 120 = \$9.90$ (e.g. sales tax).

5. Percent increase and decrease

"Increase $Q$ by $P\%$" means: take $Q$, add $P\%$ of $Q$ to it. "Decrease $Q$ by $P\%$" means: subtract.

You can do that in two steps (compute the change, then add or subtract), but there's a faster pattern: multiply by a single number. This is by far the most useful technique.

To increase by $P\%$, multiply by $1 + \tfrac{P}{100}$. To decrease by $P\%$, multiply by $1 - \tfrac{P}{100}$.

So:

• Increase $80$ by $25\%$: $80 \cdot 1.25 = 100$.
• Decrease $80$ by $25\%$: $80 \cdot 0.75 = 60$.
• Apply $8\%$ sales tax to $\$45$: $45 \cdot 1.08 = \$48.60$.
• Take $30\%$ off a $\$120$ jacket: $120 \cdot 0.70 = \$84$.

The number you multiply by is called the multiplier. It bundles "the original quantity, plus or minus the change" into one factor — and once you're thinking in multipliers, percentages become much easier to chain.

6. Percent change and the asymmetry trap

To describe a change as a percentage, the formula is:

Always divide by the old (starting) value — that's the "out of how many" the percent is measured against.

Here is the trap that catches almost everyone the first time:

A $10\%$ decrease followed by a $10\%$ increase does not return you to the original number.

Start at $\$100$. Drop $10\%$: $\$100 \cdot 0.90 = \$90$. Add $10\%$ back: $\$90 \cdot 1.10 = \$99$. You're a dollar short — and that dollar isn't a rounding error. It's because the second $10\%$ is computed off a smaller base.

In multiplier form, the round trip is $1.10 \cdot 0.90 = 0.99$, not $1$. The two percent changes are not inverses: to perfectly undo a $10\%$ decrease, you need an increase of $\tfrac{1}{0.9} - 1 \approx 11.11\%$, not $10\%$.

This asymmetry is the single most important pitfall about percentages and it shows up everywhere:

• A stock that drops $50\%$ has to double ($+100\%$) to recover.
• If your salary goes up $5\%$ in inflation and then taxes take $5\%$, you're not back to where you started.
• "Lost $40\%$ of users, then gained $40\%$" leaves the user count at $1.40 \cdot 0.60 = 0.84$ — down $16\%$ overall.

7. Working backwards from a result

Sometimes the percentage and the result are known, and the question is the starting value. Translate the situation into an equation and solve.

"A jacket is on sale for $\$84$ after a $30\%$ discount. What was the original price?"

The discount multiplier is $1 - 0.30 = 0.70$. So $\text{original} \cdot 0.70 = 84$. Solve:

The original price was $\$120$. The temptation is to "add the $30\%$ back" by computing $84 \cdot 1.30 = \$109.20$ — but that's wrong, because the $30\%$ was originally a discount off $120$, not off $84$.

The rule that prevents this mistake: to undo a multiplier, divide by it. You never undo a percent change by applying its "opposite percent." You undo it by inverting the multiplier.

8. Common pitfalls

9. Worked examples