Math Core

Every percent problem is built from the same three pieces: a whole, a part of it, and the percent that connects them. Once you can spot which piece is missing, one short equation finds it.

Three pieces, one equation

In Grade 6 you found percents of numbers with ratio tables. Now you'll use an equation that works for every percent problem, even with messy numbers.

Think about the sentence " $30\%$ of 80 is 24."

80 is the whole (the amount you're taking a percent of).
24 is the part.
$30\%$ is the percent.

Since $30\% = 0.30$ , the sentence says $0.30 \times 80 = 24$ . The percent works like a constant of proportionality: the part is always the percent (as a decimal) times the whole.

The percent equation

\text{part} = \text{percent} \times \text{whole}

Write the percent as a decimal or a fraction before you multiply or divide. A problem gives you two of the three pieces, and you solve for the third.

The word "of" usually points to the whole, and "is" usually means "equals." Translating the sentence word by word often gives you the equation directly.

Finding the part

Worked example: Find the part

A school has 640 students, and $35\%$ of them walk to school. How many students walk?

The whole is 640 and the percent is $35\% = 0.35$ . The part is missing.

\text{part} = 0.35 \times 640 = 224

224 students walk to school.

Finding the percent

When the percent is missing, divide the part by the whole. Then change the decimal to a percent by multiplying by 100.

Worked example: Find the percent

Marco made 18 of his 24 free throws. What percent did he make?

The part is 18 and the whole is 24. Let $p$ be the percent as a decimal.

\begin{aligned} 18 &= p \times 24 \\ p &= \frac{18}{24} = 0.75 \end{aligned}

$0.75 = 75\%$ . Marco made 75% of his free throws.

Finding the whole

When the whole is missing, call it $w$ and solve. You divide the part by the percent.

Worked example: Find the whole

Priya has read 150 pages of a novel. That is $40\%$ of the book. How many pages long is the book?

The part is 150, the percent is $0.40$ , and the whole is missing.

\begin{aligned} 150 &= 0.40 \times w \\ w &= \frac{150}{0.40} = 375 \end{aligned}

The book has 375 pages. Check: $0.40 \times 375 = 150$ . Correct.

Percents less than 1% and more than 100%

Percents are not stuck between 1 and 100.

A percent greater than 100% gives a part bigger than the whole. $150\%$ of 60 is $1.5 \times 60 = 90$ .
A percent less than 1% gives a tiny part. $0.5\%$ of 600 is $0.005 \times 600 = 3$ .

Worked example: An unusual percent

A town had 2,000 residents ten years ago. Today its population is $135\%$ of that. How many residents does it have today?

$135\% = 1.35$ , so the population is $1.35 \times 2{,}000 = 2{,}700$ residents. It makes sense that the answer is more than 2,000, because the percent is more than $100\%$ .

Common mistake

Be careful when you change a percent to a decimal. $8\% = 0.08$ , not $0.8$ . And $0.5\% = 0.005$ , not $0.5$ . Always move the decimal point two places left, adding zeros if you need to.

Tip

Estimate first. If $40\%$ of a book is 150 pages, the whole book must be more than 150 pages. If your answer is smaller than the part (and the percent is under $100\%$ ), you probably multiplied when you should have divided.