How to Find a Percent of a Number

Part = percent × whole. The word "of" between a percent and a number means multiply. A jacket at $25\%$ off, a tip at $18\%$, a poll where $60\%$ agree, all of them come down to that one multiplication.

Problem

A wildlife reserve counts $200$ nesting birds, and $25\%$ are herons. How many herons are there?

Show a hint

The word "of" means multiply. Rewrite $25\%$ as $\tfrac{1}{4}$ or $0.25$, then multiply it by $200$.
One quarter of $200$ is $200 \div 4 = 50$. Or $0.25 \times 200 = 50$.

Show the full solution

$25\% = \tfrac{1}{4}$, and "of" means multiply, so $$25\% \text{ of } 200 = \tfrac{1}{4} \times 200 = 50.$$ There are $\boxed{50}$ herons. Decimals work just as well, $0.25 \times 200 = 50$.

Problem

A bakery sold $35$ loaves, and $60\%$ were sourdough. How many sourdough loaves were sold?

Show a hint

Rewrite $60\%$ as $0.6$ (or $\tfrac{3}{5}$), then multiply by $35$.
$0.6 \times 35 = 21$. Or $\tfrac{3}{5} \times 35 = 3 \times 7 = 21$.

Show the full solution

Convert and multiply. Since $60\% = \tfrac{3}{5} = 0.6$, $$60\% \text{ of } 35 = \tfrac{3}{5} \times 35 = 3 \times 7 = 21.$$ So $\boxed{21}$ sourdough loaves were sold.

Problem

A startup had $15$ employees and grew to $300\%$ of that size. How many employees are there now?

Show a hint

A percent over $100$ becomes a number bigger than $1$. Here $300\% = 3$.
Multiply $3 \times 15 = 45$.

Show the full solution

$300\% = \tfrac{300}{100} = 3$, so $$300\% \text{ of } 15 = 3 \times 15 = 45.$$ There are $\boxed{45}$ employees. A percent above $100$ gives more than the whole, so an answer larger than $15$ is the right sign.

Finding $25\%$ of $200$. Split the whole bar of $200$ into four equal quarters of $50$. One quarter is $25\%$, so $25\%$ of $200$ is one of those quarters, $\tfrac{1}{4} \times 200 = 50$. Taking a percent of a number is multiplying the whole by that fraction.

Problem

A theater holds $64$ seats, and $12\tfrac{1}{2}\%$ are reserved for staff. Since $12\tfrac{1}{2}\% = \tfrac{1}{8}$, how many seats are reserved?

Show a hint

First convert. $12\tfrac{1}{2}\% = 12.5\% = \tfrac{12.5}{100} = \tfrac{1}{8}$.
One eighth of $64$ is $64 \div 8 = 8$.

Show the full solution

Convert the fractional percent to a clean fraction, $12\tfrac{1}{2}\% = \tfrac{12.5}{100} = \tfrac{1}{8}$. Then $$12\tfrac{1}{2}\% \text{ of } 64 = \tfrac{1}{8} \times 64 = 8.$$ So $\boxed{8}$ seats are reserved.

Problem

A water sample weighs $80$ g, and $2.5\%$ is dissolved minerals. How many grams of minerals are there?

Show a hint

Slide the point two places left, so $2.5\% = 0.025$.
$0.025 \times 80 = 2$.

Show the full solution

Convert the percent to a decimal by sliding the point two places left, $2.5\% = 0.025$. Then $$2.5\% \text{ of } 80 = 0.025 \times 80 = 2.$$ So there are $\boxed{2}$ grams of dissolved minerals.

Problem

A class read $30$ books out of a goal of $120$. What percent of the goal did they reach? Give just the number.

Show a hint

Make the fraction part over whole, $\tfrac{30}{120}$, and simplify.
$\tfrac{30}{120} = \tfrac{1}{4} = 0.25$, and $0.25$ as a percent is $25\%$.

Show the full solution

Divide the part by the whole, then write it as a percent. $$\tfrac{30}{120} = \tfrac{1}{4} = 0.25 = 25\%.$$ So $30$ is $\boxed{25}$ percent of $120$.

Problem

A phone battery holds $80$ units full and $60$ units now. What percent of $80$ is $60$? Give just the number.

Show a hint

The part is $60$ and the whole is $80$, so form $\tfrac{60}{80}$ and simplify.
$\tfrac{60}{80} = \tfrac{3}{4} = 0.75 = 75\%$.

Show the full solution

Divide the part by the whole, then convert to a percent. $$\tfrac{60}{80} = \tfrac{3}{4} = 0.75 = 75\%.$$ So $60$ is $\boxed{75}$ percent of $80$.

Problem

A shop bought a desk for $\$17$ and resold it for $\$51$. What percent is $51$ of $17$? Give just the number.

Show a hint

Form $\tfrac{51}{17}$. Since $51$ is bigger than $17$, expect a percent above $100$.
$\tfrac{51}{17} = 3 = 300\%$.

Show the full solution

Divide the part by the whole, $\tfrac{51}{17} = 3$, and $3 = 300\%$. So $51$ is $\boxed{300}$ percent of $17$. The resale price is three times the cost, which is what a percent above $100$ always means.

The three percent questions are one relationship, $\text{part} = \text{percent} \times \text{whole}$, read from three corners. Cover the part and you multiply percent by whole. Cover the percent and you divide the part by the whole. Cover the whole and you divide the part by the percent.

Problem

A scholarship covers $\$80$ of a fee, and that is exactly $20\%$ of the full fee. $80$ is $20\%$ of what number?

Show a hint

The relationship is part equals percent times whole, so whole equals part divided by percent. Use $20\% = 0.2$.
$80 \div 0.2 = 400$. Check, $20\%$ of $400$ is $80$.

Show the full solution

With $20\% = 0.2$, divide the part by the percent, $$\text{whole} = \frac{80}{0.2} = 400.$$ So $80$ is $20\%$ of $\boxed{400}$. Check by going forward, $0.2 \times 400 = 80$.

Problem

A hiker has walked $27$ km, which is $75\%$ of the full trail. $27$ is $75\%$ of what number?

Show a hint

Whole equals part divided by percent, so compute $27 \div 0.75$.
$27 \div \tfrac{3}{4} = 27 \times \tfrac{4}{3} = 36$. Check, $75\%$ of $36$ is $27$.

Show the full solution

Divide the part by the percent. With $75\% = \tfrac{3}{4}$, $$\text{whole} = 27 \div \tfrac{3}{4} = 27 \times \tfrac{4}{3} = 36.$$ The trail is $\boxed{36}$ kilometers long. Check by going forward, $\tfrac{3}{4} \times 36 = 27$.

Problem

A startup's revenue this month is $\$18$k, which is $150\%$ of last month's. $18$ is $150\%$ of what number?

Show a hint

Whole equals part divided by percent, so compute $18 \div 1.5$.
$18 \div 1.5 = 12$. Since the percent is over $100$, the whole is smaller than the part.

Show the full solution

Divide the part by the percent, with $150\% = 1.5$. $$\text{whole} = \frac{18}{1.5} = 12.$$ Last month's revenue was $\boxed{12}$ thousand dollars. Since $150\%$ is more than $100\%$, the whole comes out smaller than the part, and $12$ is indeed under $18$.

Problem

A museum finds $30\%$ of $80$ visitors came by bus. What percent of $120$ school visitors is that same bus count? Give just the number.

Show a hint

Step one, find $30\%$ of $80$. That is $0.3 \times 80 = 24$.
Step two, what percent is $24$ of $120$? Form $\tfrac{24}{120} = 0.2 = 20\%$.

Show the full solution

Do it in two steps. First the part, $30\%$ of $80 = 0.3 \times 80 = 24$. Then ask what percent $24$ is of $120$, $$\tfrac{24}{120} = \tfrac{1}{5} = 0.2 = 20\%.$$ So $30\%$ of $80$ is $\boxed{20}$ percent of $120$.

Practice these ideas

Practice

A farmers market had $90$ stalls, and $40\%$ sold vegetables. How many stalls sold vegetables?

Show the solution

$40\% = 0.4$, so $40\% \text{ of } 90 = 0.4 \times 90 = 36$. So $\boxed{36}$ stalls sold vegetables.

Practice

A song had $12$ thousand listeners last week and climbed to $250\%$ of that this week. How many thousand listeners this week?

Show the solution

$250\% = \tfrac{250}{100} = 2.5$, so $250\% \text{ of } 12 = 2.5 \times 12 = 30$. So there are $\boxed{30}$ thousand listeners.

Practice

A puzzle has $16$ pieces, and $37\tfrac{1}{2}\%$ are edge pieces. How many edge pieces are there?

Show the solution

$37\tfrac{1}{2}\% = \tfrac{37.5}{100} = \tfrac{3}{8}$, so $\tfrac{3}{8} \times 16 = 6$. So there are $\boxed{6}$ edge pieces.

Practice

A reservoir holds $150$ million liters, and $8\%$ is in reserve. How many million liters are in reserve?

Show the solution

$8\% = 0.08$, so $8\% \text{ of } 150 = 0.08 \times 150 = 12$. So $\boxed{12}$ million liters are in reserve.

Practice

A team won $18$ of its $72$ games. What percent did the team win? Give just the number.

Show the solution

$\tfrac{18}{72} = \tfrac{1}{4} = 0.25 = 25\%$. So the team won $\boxed{25}$ percent of its games.

Practice

A jar holds $40$ marbles and $30$ are blue. What percent of $40$ is $30$? Give just the number.

Show the solution

$\tfrac{30}{40} = \tfrac{3}{4} = 0.75 = 75\%$. So $30$ is $\boxed{75}$ percent of $40$.

Practice

A sequel sold $84$ thousand copies. The original sold $56$ thousand. What percent is $84$ of $56$? Give just the number.

Show the solution

$\tfrac{84}{56} = \tfrac{3}{2} = 1.5 = 150\%$. So $84$ is $\boxed{150}$ percent of $56$.

Practice

A deposit of $\$21$ is $30\%$ of a bike's total price. $21$ is $30\%$ of what number?

Show the solution

With $30\% = 0.3$, divide the part by the percent, $\tfrac{21}{0.3} = 70$. The price is $\boxed{70}$ dollars. Check by going forward, $0.3 \times 70 = 21$.

Practice

A runner has finished $45$ laps, which is $90\%$ of the race. $45$ is $90\%$ of what number?

Show the solution

With $90\% = 0.9$, divide the part by the percent, $\tfrac{45}{0.9} = 50$. The race is $\boxed{50}$ laps. Check by going forward, $0.9 \times 50 = 45$.

Practice

This year a town recorded $120$ building permits, which is $150\%$ of last year's. $120$ is $150\%$ of what number?

Show the solution

With $150\% = 1.5$, divide the part by the percent, $\tfrac{120}{1.5} = 80$. Last year had $\boxed{80}$ permits. Check by going forward, $1.5 \times 80 = 120$.

Practice

A tasting portion is $5$ g, which is $12\tfrac{1}{2}\%$ of a full serving. $5$ is $12\tfrac{1}{2}\%$ of what number?

Show the solution

$12\tfrac{1}{2}\% = \tfrac{1}{8}$, and the whole is the part divided by the percent, so $5 \div \tfrac{1}{8} = 5 \times 8 = 40$. A full serving is $\boxed{40}$ grams. Check by going forward, $\tfrac{1}{8} \times 40 = 5$.

Practice

Chain two steps. Take $60\%$ of $500$, then take $20\%$ of that result. What is $20\%$ of $60\%$ of $500$?

Show the solution

Work from the inside out. First $60\%$ of $500 = 0.6 \times 500 = 300$. Then $20\%$ of $300 = 0.2 \times 300 = 60$. So $20\%$ of $60\%$ of $500$ is $\boxed{60}$.

A Percent of a Number

Practice these ideas