Maths SolverThe word percent means “per hundred.” A percentage is just a fraction with a denominator of $100$ written in a different notation: $25\% = \tfrac{25}{100} = 0.25$. Once you have that conversion in mind, every percentage problem reduces to one of three simple multiplication or division questions.
The trouble students have with percentages is almost never the arithmetic. It is keeping straight which of the three questions is being asked. So that is what this page is about.
The three question types
Type 1 — what is X% of Y?
This is the simplest. Convert the percentage to a decimal and multiply. “What is $20\%$ of $150$?” gives $0.20 \times 150 = 30$. The general formula is
$$\text{result} = \frac{X}{100} \times Y.$$
Type 2 — X is what percent of Y?
This one is asking for the percentage itself. Divide $X$ by $Y$ and multiply by $100$. “$30$ is what percent of $150$?” gives $\tfrac{30}{150} \times 100 = 20\%$.
$$P = \frac{X}{Y} \times 100.$$
Type 3 — X is P% of what?
This is the “reverse percentage” type and the one that catches people. You know what the result is and what percentage it represents; you have to find the original. “$30$ is $20\%$ of what?” gives $30 \div 0.20 = 150$.
$$Y = \frac{X}{P / 100} = \frac{100 X}{P}.$$
Percentage change — the fourth case
A close cousin of the three above is “by what percent did $X$ change?” If a price went from $80$ to $100$, the percentage increase is the change ($100 - 80 = 20$) divided by the original ($80$), times $100$:
$$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%.$$
$\tfrac{20}{80} \times 100 = 25\%$ increase. Note the denominator is the original value, not the new one. This is the single most common mistake in percentage-change problems.
The mistakes I see most often
1. Dividing by the wrong number on percentage change
“The price rose from $80$ to $100$” is a $25\%$ increase. “The price fell from $100$ to $80$” is a $20\%$ decrease. The two changes are the same in absolute terms ($20$) but different in relative terms because the base is different. Always divide by the starting value.
2. Confusing “percent of” with “percent more than”
“$120\%$ of $50$” is $60$. “$120\%$ more than $50$” is $50 + 60 = 110$. The first is a multiplication; the second is a multiplication followed by an addition.
3. Adding percentages directly when they apply to different bases
A $10\%$ increase followed by a $10\%$ decrease does not get you back where you started. It gets you to $1.10 \times 0.90 = 0.99$ times the original, i.e. a $1\%$ decrease overall. Percentages compound; they do not add.
4. The “reverse percentage” trap
“After a $25\%$ tax was added, the bill was $\pounds 50$. What was the price before tax?” The instinct is to take $25\%$ off $\pounds 50$, getting $\pounds 37.50$. That is wrong. The $25\%$ was added to the original, so $\pounds 50$ is $125\%$ of what we want. Divide: $50 \div 1.25 = \pounds 40$. Always solve reverse-percentage problems by setting up an equation, not by symmetry.
Where this comes up later
Percentages appear in finance (interest, inflation, taxes, discounts), in science (percent error, percent yield, concentration), and in statistics (probability, polls, confidence intervals). Almost any real-world question with a number in it can be re-asked as a percentage question.
References
- Wikipedia: Percentage, Percentage change.
- Khan Academy: Intro to percentages.