What is a percentage?

A percentage (p%) is a hundredth of a whole: p% = p/100. For example 25% is 25/100, that is 1/4 of the whole. The % sign just replaces writing "over one hundred".

How do you find a percentage of a number?

Multiply the number by the fraction the percentage represents: p% of c is (p/100) · c. For example 20% of 150 is (20/100) · 150 = 30.

How do you compute a percentage change?

Subtract the old value from the new one, divide by the old value, and multiply by 100%: Δ% = (new − old) / old · 100%. A positive result is an increase, a negative one a decrease. A price rising from 80 to 100 changes by (100 − 80)/80 · 100% = 25%.

How do you write an increase or a discount as a single multiplier?

A rise of p% is a multiplication by 1 + p/100 and a discount of p% is a multiplication by 1 − p/100. A price 15% higher is 1.15 times the old one, a price 15% lower is 0.85 times it. The notation is shorter than "work out the percentage, then add it" and, above all, it combines: several changes in a row are the product of their multipliers.

Does a 20% increase followed by a 20% decrease return to the starting point?

No. A percentage is always taken from a DIFFERENT base: a 20% increase on 100 gives 120, and a 20% decrease on 120 gives 96, not 100 — because the second percentage is taken from the larger number.

Intermediate level

Percentages

Q: How do you recover the whole when you only know a part and its percentage?

Divide the known part by the fraction the percentage represents. If 30 is 20% of some number, the whole is 30 ÷ (20/100) = 30 ÷ 0.2 = 150.

A percentage is a hundredth of a whole. Learn percentage notation, finding a percentage of a number, recovering the whole from a known part, computing percentage change, and writing an increase or a discount as a single multiplier.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

FractionsA fraction writes part of a whole as a numerator and a denominator. Learn equivalent fractions, all four operations on fractions, mixed numbers, comparing fractions and reducing to lowest terms.
DecimalsA decimal writes part of a whole with a decimal point instead of a fraction bar. Learn place value, converting both ways between a fraction and a decimal, all four operations, rounding and repeating expansions.

Where this is used

Real situations where you count exactly the way this lesson teaches:

The sales rack
A 349 jacket at 30% off costs 349 · 0.7 = 244.30, and the promised "another 20% at the till" is not a 50% discount but 244.30 · 0.8 = 195.44 — 44% off altogether.
See the lesson: Decimals
Savings
A 12,000 deposit paying 2.8% a year earns 336 in interest, but 19% of that goes in tax, so 272.16 reaches the account — an effective 2.27% a year.
Retail
A shopkeeper buys at 18 and sells at 30. The same 12 of profit is a 66.7% markup measured against the cost or a 40% margin measured against the selling price — all that changes is which figure counts as the whole.

All formulas

Percentage notation
$p\% = \frac{p}{100}$
a percentage is a hundredth of a whole
Percentage of a number
$p\% \text{ of } c = \frac{p}{100} \cdot c$
how much p percent of the whole c is
Whole from a known part
$c = \text{part} \div \frac{p}{100}$
given the part and its percentage, recover the whole
Percentage change
$\Delta\% = \frac{\text{new} - \text{old}}{\text{old}} \cdot 100\%$
positive is an increase, negative is a decrease
Change multiplier
$\text{new} = \text{old} \cdot \left(1 \pm \frac{p}{100}\right)$
a rise of p% multiplies by 1 + p/100, a discount by 1 − p/100
Two changes in a row
$\text{new} = \text{old} \cdot \left(1 + \frac{p_1}{100}\right)\left(1 + \frac{p_2}{100}\right)$
multiply the multipliers; never add the percentages

A percentage is a hundredth of a whole — rather than writing a fraction with a fixed denominator of $100$ , we use the $\%$ sign:

p\% = \frac{p}{100}

For example:

25\% = \frac{25}{100} = \frac{1}{4}

A percentage is simply another way of writing a fraction whose denominator is always $100$ .

Percentage of a number

To find how much $p$ percent of a number $c$ is, multiply $c$ by the fraction $\frac{p}{100}$ :

p\% \text{ of } c = \frac{p}{100} \cdot c

For example $20\%$ of $150$ is:

\frac{20}{100} \cdot 150 = 0.2 \cdot 150 = 30

What is 15% of 80?

Whole from a known part

Sometimes you only know a part and what percentage of the whole it represents, and you want the whole. Divide the part by the fraction the percentage represents:

If $30$ is $20\%$ of some number, the whole is:

30 \div \frac{20}{100} = 30 \div 0.2 = 150

45 is 30% of what number?

Percentage change

Percentage change tells you how much a value increased or decreased relative to its starting point:

\Delta\% = \frac{\text{new} - \text{old}}{\text{old}} \cdot 100\%

A positive result means an increase, a negative one a decrease. A price rising from $80$ to $100$ changes by $\frac{100 - 80}{80} \cdot 100\% = 25\%$ .

Watch out for a common trap: a $20\%$ increase followed by a $20\%$ decrease does not return to the starting point — each percentage is taken from a different (current) base. A $20\%$ increase on $100$ gives $120$ ; a $20\%$ decrease on $120$ gives $96$ , not $100$ .

The change multiplier

An increase or a discount can be worked out in two steps ("find $15\%$ , then add or subtract it") or in one — by multiplying straight away by the change multiplier:

\text{new} = \text{old} \cdot \left(1 \pm \frac{p}{100}\right)

The $1$ is the whole you keep and $\frac{p}{100}$ is the part you add or take away. A $15\%$ rise is therefore a multiplication by $1.15$ , and a $15\%$ discount a multiplication by $0.85$ :

\text{price} \cdot 1.15 \qquad \text{price} \cdot 0.85

A multiplier also reads backwards: $0.7$ means " $70\%$ of the price", that is a $30\%$ discount, and $1.08$ is an $8\%$ rise. A price of $349$ marked down by $30\%$ is one multiplication, $349 \cdot 0.7 = 244.30$ , rather than a discount worked out and subtracted separately.

The real gain shows up with several changes in a row. Percentages must not be added there — multipliers may be multiplied:

\text{new} = \text{old} \cdot \left(1 + \frac{p_1}{100}\right)\left(1 + \frac{p_2}{100}\right)

This is the one-line explanation of the trap in the previous section: a $20\%$ rise followed by a $20\%$ fall gives

1.2 \cdot 0.8 = 0.96

that is $96\%$ of the starting value — a $4\%$ drop, not a return to where you began. The product of multipliers is itself a multiplier, so the combined result reads off immediately: $30\%$ off and then another $20\%$ at the till is $0.7 \cdot 0.8 = 0.56$ , a combined discount of $44\%$ rather than $50\%$ .

A price rose by 25% and then fell by 12%. What is the overall change?

The same multiplier raised to a power describes a change repeated many times over — that is what compound interest is built on, and it has its own lesson, together with percentage points.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

25% × 24 =

Common mistakes

Confusing "percentage of a number" with "what percentage one number is of another" — these are two different questions: one multiplies, the other divides.
Computing percentage change from the new value instead of the old one — the denominator is always the starting value, never the ending one.
Assuming an increase and a decrease of the same percentage cancel out — they do not, because the base the percentage is taken from changes between the two steps.
Adding percentages instead of multiplying multipliers — $30\%$ off and another $20\%$ off is $0.7 \cdot 0.8 = 0.56$ , that is $44\%$ , not $50\%$ .
Confusing a multiplier with a percentage — $0.85$ does not mean " $85\%$ off", it means " $85\%$ of the price", that is $15\%$ off.

Formula card

Topic: Percentages

Percentage notation
$p\% = \frac{p}{100}$
a percentage is a hundredth of a whole
Percentage of a number
$p\% \text{ of } c = \frac{p}{100} \cdot c$
how much p percent of the whole c is
Whole from a known part
$c = \text{part} \div \frac{p}{100}$
given the part and its percentage, recover the whole
Percentage change
$\Delta\% = \frac{\text{new} - \text{old}}{\text{old}} \cdot 100\%$
positive is an increase, negative is a decrease
Change multiplier
$\text{new} = \text{old} \cdot \left(1 \pm \frac{p}{100}\right)$
a rise of p% multiplies by 1 + p/100, a discount by 1 − p/100
Two changes in a row
$\text{new} = \text{old} \cdot \left(1 + \frac{p_1}{100}\right)\left(1 + \frac{p_2}{100}\right)$
multiply the multipliers; never add the percentages

Frequently asked questions

$A fraction bar and a percent sign rendered as geometric forms, with a circle split into equal orbital slices, against a cosmic nebula in violet and magenta$

Numbers

A percentage is a fraction in disguise — three ways to write one number

1/2, 0.5 and 50% are not three different things but three ways of writing the same number. Here is where the fraction bar, the decimal point and the % sign came from, why 1/3 never ends, why +20% and −20% do not return you to the start, and why fractions are the first serious wall in school mathematics.

11 min