Math Core

A store sign says "25% off." A weather app says "60% chance of rain." A test score says "90%." A percent is a special kind of ratio that compares a number to 100, which makes different amounts easy to compare.

What a percent means

The word percent means "per hundred." The symbol is %.

Definition

Percent

A percent is a rate per 100. So $p\%$ means $p$ out of 100:

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

For example, $37\%$ means 37 out of 100, or $\dfrac{37}{100}$ , or $0.37$ .

Picture a grid of 100 squares. If 45 squares are shaded, then $45\%$ of the grid is shaded. The whole grid is $100\%$ .

Fractions, decimals and percents

Since a percent is "out of 100," you can switch between fractions, decimals and percents.

Percent to decimal: divide by 100. Move the decimal point two places left. $62\% = 0.62$ , and $8\% = 0.08$ .
Decimal to percent: multiply by 100. Move the decimal point two places right. $0.3 = 30\%$ .
Fraction to percent: write an equivalent fraction with a denominator of 100.

Worked example: Rewriting as a percent

Write each number as a percent.

$\dfrac{3}{4}$
$\dfrac{7}{20}$
$0.05$

Solutions.

$\dfrac{3}{4} = \dfrac{3 \times 25}{4 \times 25} = \dfrac{75}{100} = 75\%$
$\dfrac{7}{20} = \dfrac{7 \times 5}{20 \times 5} = \dfrac{35}{100} = 35\%$
$0.05 = \dfrac{5}{100} = 5\%$

Some percents are worth knowing by heart:

Percent	10%	20%	25%	50%	75%	100%
Fraction	$\frac{1}{10}$	$\frac{1}{5}$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{3}{4}$	$1$

Finding a percent of a number

A percent is a ratio, so you can use a ratio table. Think of "out of 100" as the first column.

Worked example: Percent of a number

What is $30\%$ of 60?

$30\%$ means 30 for every 100. Find the number that goes with 60.

Part	30	3	18
Whole	100	10	60

Divide by 10, then multiply by 6. So $30\%$ of 60 is 18.

You can also multiply by the decimal: $0.30 \times 60 = 18$ .

Part, whole and percent

\text{part} = \text{percent (as a decimal)} \times \text{whole}

If you know any two of these, you can find the third. A ratio table works for all three kinds of questions.

Tip

To find $10\%$ of a number, divide by 10. Then build other percents from it. For example, $10\%$ of 80 is 8, so $5\%$ is 4, $20\%$ is 16, and $15\%$ is $8 + 4 = 12$ .

Finding the whole

Sometimes you know the part and the percent, and you need the whole.

Worked example: Finding the whole

Leah has saved $24. That is $40\%$ of the cost of a new skateboard. How much does the skateboard cost?

The part is 24, and the percent is 40. The whole is unknown.

Part (dollars)	40	4	24
Whole (dollars)	100	10	60

Divide by 10 to get 4 for every 10, then multiply by 6 to get 24. The skateboard costs $60.

Check: $40\%$ of 60 is $0.4 \times 60 = 24$ . ✓

Common mistake

Watch out for which number is the whole. In "24 is $40\%$ of what number?", 24 is the part, not the whole. A common mistake is to find $40\%$ of 24 instead. Ask yourself: should the answer be bigger or smaller than 24? Since 24 is only $40\%$ of the whole, the whole must be bigger.

Finding the percent

Worked example: What percent?

Ravi answered 42 out of 50 questions correctly. What percent did he get right?

Write the ratio as a fraction out of 100:

\frac{42}{50} = \frac{42 \times 2}{50 \times 2} = \frac{84}{100} = 84\%

Ravi got 84% right.