Percents

Percent means "per hundred". The word comes from the Latin per centum, "by the hundred", and the sign % stands for "out of 100". So 25% means 25 out of every 100. One percent is one hundredth, ${1\%=\frac{1}{100}=0.01}$, and 25% of something is $\frac{25}{100}$ of it.

25 of the 100 small squares are shaded: ${25\%}={\frac{25}{100}}={\frac{1}{4}}$

Suppose there are 4 people in a room. Then 100% of them means all 4 people, 50% means half of them (2 people), 25% means a quarter (1 person) and 0% means nobody. If 4 more people come in, there are 8 people in the room, which is 200% of the original number. A percent can be more than 100%.

On this page you will learn how to change percents into decimals and fractions and how to solve the three basic percent problems: finding a percent of a number, finding what percent one number is of another, and finding the whole from a part. Then come percent increase and decrease, reverse percentages and percent changes that follow one another. If you only need a quick answer, use the percentage calculator.

Percents, Decimals and Fractions

A percent, a decimal and a fraction are three ways to write the same number: ${50\%}={0.5}={\frac{1}{2}}$.

Some percents come up so often that it pays to know them by heart:

Percent	Decimal	Fraction
$1\%$	$0.01$	$\frac{1}{100}$
$5\%$	$0.05$	$\frac{1}{20}$
$10\%$	$0.1$	$\frac{1}{10}$
$12.5\%$	$0.125$	$\frac{1}{8}$
$20\%$	$0.2$	$\frac{1}{5}$
$25\%$	$0.25$	$\frac{1}{4}$
$33\tfrac{1}{3}\%$	$0.333\ldots$	$\frac{1}{3}$
$50\%$	$0.5$	$\frac{1}{2}$
$75\%$	$0.75$	$\frac{3}{4}$
$100\%$	$1$	$1$
$150\%$	$1.5$	$\frac{3}{2}$

Careful with small percents: ${8\%=0.08}$, not $0.8$. And ${0.5\%=0.005}$ is half of one percent, not one half: one half is 50%.

Finding a Percent of a Number

Example 1. Find 20% of 40.

${20\%\ \text{of}\ 40}={0.2\times 40}=8$

The word "of" tells you to multiply.

Example 2. A school has 600 students, and 45% of them come to school by bus. How many students is that?

${0.45\times 600}=270$

Example 3. A percent can be more than 100% or less than 1%:

${120\%\ \text{of}\ 50}={1.2\times 50}=60$

${0.5\%\ \text{of}\ 3000}={0.005\times 3000}=15$

Example 4. Cindy needs 8 m of garden hose. The store has a 30 m reel, but a label on it says that 60% of the hose has already been sold. Is there enough left for Cindy?

Solution: ${{0.6\times 30}=18}$ m have been sold, so ${{30-18}=12}$ m are left. That is enough. Faster: ${{100\%-60\%}={40\%}}$ of the reel is left, and ${{0.4\times 30}=12}$ m.

Percents in Your Head

• 10%: divide by 10. 10% of 360 is 36.
• 5%: take half of 10%. 5% of 360 is 18.
• 1%: divide by 100. 1% of 360 is 3.6.
• 50%, 25% and 20% are a half, a quarter and a fifth. 25% of 360 is 90.
• Build other percents from these: 15% is 10% plus 5%, so 15% of 360 is ${{36+18}=54}$.
• Swap the numbers: $p\%$ of $a$ is the same as $a\%$ of $p$, because ${\frac{p}{100}\times a}={\frac{a}{100}\times p}$. 8% of 50 is hard to work out in your head, but 50% of 8 is easy. Both are 4.

For example, a 15% tip on a $\$36$ bill: 10% is $\$3.60$ and 5% is $\$1.80$, so the tip is $\$5.40$.

What Percent Is One Number of Another?

Example 1. 80 is what percent of 160?

${\frac{80}{160}\times 100\%}={0.5\times 100\%}={50\%}$

Example 2. You scored 45 points out of 60 on a test. What percent is that?

${\frac{45}{60}\times 100\%}={0.75\times 100\%}={75\%}$

Example 3. A quiz has 5 questions: three are worth 3 marks each, and two are worth 4 marks each. You answered two of the 3-mark questions and one of the 4-mark questions correctly. What percent of the marks did you get?

Solution: The quiz is worth ${{3\times 3+2\times 4}=17}$ marks, and you got ${{2\times 3+4}=10}$ marks:

${\frac{10}{17}\times 100\%}\approx{58.8\%}$

The division doesn't come out exact, so the answer is rounded, and we write ≈ ("approximately equal to") instead of =.

Example 4. Ryan collects sports cards: he has 32 baseball cards, 25 football cards and 47 basketball cards. What percent of his collection is each sport?

Solution: The whole is the number of all the cards, ${{32+25+47}=104}$.

Baseball: ${\frac{32}{104}\times 100\%}\approx{30.8\%}$

Football: ${\frac{25}{104}\times 100\%}\approx{24.0\%}$

Basketball: ${\frac{47}{104}\times 100\%}\approx{45.2\%}$

Together the three parts make up the whole collection, so their percents add up to 100%. Rounded percents can add up to a little more or a little less, such as 99.9% or 100.1%.

Example 5. The part can be bigger than the whole. 45 is what percent of 40?

${\frac{45}{40}\times 100\%}={1.125\times 100\%}={112.5\%}$

Finding the Whole from a Part

Example 1. 12 is 30% of what number?

${12\div 0.3}=40$

Check: ${{0.3\times 40}=12}$ ✓

Example 2. 6 students in a class wear glasses. That is 20% of the class. How many students are in the class?

${6\div 0.2}=30$

You can also go through 10%: if 20% of the class is 6 students, then 10% is 3 students, and 100% is ${{10\times 3}=30}$ students.

Example 3. You have saved $\$150$. That is 60% of the price of a bike. How much does the bike cost?

${150\div 0.6}={\$250}$

One Formula for All Three Problems

All three problems come from one formula that connects the part, the percent and the whole:

You know two of the three numbers and look for the third one. The three questions below use the same numbers, because 8 is 20% of 40:

Question	Solution
What is 20% of 40? (the part)	${0.2\times 40}=8$
8 is what percent of 40? (the percent)	${\frac{8}{40}\times 100\%}={20\%}$
8 is 20% of what number? (the whole)	${8\div 0.2}=40$

Percent Increase and Decrease

When a quantity changes, the percent change compares the change with the original (old) value:

Example 1. You had 80 stamps and collected more until you had 120. By what percent did your collection grow?

${\frac{120-80}{80}\times 100\%}={\frac{40}{80}\times 100\%}={50\%}$

Example 2. Then you traded some of your 120 stamps for a friend's game, and 100 stamps were left. By what percent did the number of stamps go down?

${\frac{100-120}{120}\times 100\%}={-\frac{20}{120}\times 100\%}\approx{-16.7\%}$

The number of stamps went down by about 16.7%. The change of 20 stamps is compared with 120, the number you had before the change.

Example 3. The population of a town grew from 12,000 to 12,600. That is 600 people more:

${\frac{600}{12{,}000}\times 100\%}={5\%}$

Increasing and Decreasing by a Percent

When you increase a number by $p\%$, you add $p\%$ of it to it, so you get ${100\%+p\%}$ of the number. When you decrease it by $p\%$, you get ${100\%-p\%}$ of it. So you can do it in one step, by multiplying by a multiplier:

Example 4. A video game used to cost $\$40$. Then its price went up by 20%. What is the new price?

In two steps: the price went up by ${{0.2\times 40}={\$8}}$, so the new price is ${{40+8}={\$48}}$. In one step: ${{40\times 1.2}={\$48}}$.

Example 5. A jacket costs $\$250$. In a sale it is 30% off, so the sale price is ${{250\times 0.7}={\$175}}$.

"More Than" and "Less Than"

The percent is always taken of the number you compare with, the one after "than". That's why both of these statements are true:

• 50 is 25% more than 40, because ${\frac{10}{40}}={25\%}$;
• 40 is 20% less than 50, because ${\frac{10}{50}}={20\%}$.

The difference is 10 in both cases, but it is compared with different numbers.

Reverse Percentages: Finding the Original Value

Sometimes you know the value after a percent change and need the value before it. Then divide by the multiplier:

Example 1. After a 20% discount, a jacket costs $\$64$. What was the price before the discount?

Solution: The sale price is ${{100\%-20\%}={80\%}}$ of the original price, so

${\text{original price}}={64\div 0.8}={\$80}$

Check: 20% of $\$80$ is $\$16$, and ${{80-16}=64}$ ✓. A bar model shows the same thing:

80% of the price is $\$64$, so 10% is $\$8$ and 100% is $\$80$.

Careful: don't add 20% of the sale price. ${{64+0.2\times 64}={\$76.80}}$ is wrong, because the discount was 20% of the original price, not 20% of $\$64$.

Example 2. A price with 8% sales tax is $\$54$. The price before tax was ${{54\div 1.08}={\$50}}$.

Example 3. After a 25% raise, a salary is $\$2{,}500$ a month. Before the raise it was ${{2{,}500\div 1.25}={\$2{,}000}}$.

One Percent Change After Another

When one percent change follows another, the second percent is taken of the new value, not of the original one. So you can't just add the percents. Multiply the multipliers instead.

Example 1. A price goes up by 10% and then down by 10%. It doesn't come back to where it started:

${1.1\times 0.9}={0.99}$

The final price is 99% of the original price, so it is 1% lower. For example, $\$100$ becomes $\$110$ and then $\$99$.

Example 2. A 20% discount followed by another 10% off is not 30% off:

${0.8\times 0.9}={0.72}$

You pay 72% of the price, so the total discount is 28%.

Example 3. A share price falls by 50%. To get back to where it was, it has to rise by 100%, because ${{0.5\times 2}=1}$.

Example 4. A town grows by 5% a year for 3 years:

${1.05\times 1.05\times 1.05}={1.05^3}\approx{1.158}$

In total the town grows by about 15.8%, not 15%, because each year the 5% is taken of a bigger number. Savings that earn compound interest grow the same way. The compound interest calculator below does these calculations for you.

Percent or Percentage Points?

When a quantity that is itself a percent changes, there are two ways to describe the change. Suppose a bank raises its interest rate from 4% to 5%:

• the rate went up by 1 percentage point, because ${{5-4}=1}$;
• the rate went up by 25%, because ${\frac{5-4}{4}\times 100\%}={25\%}$.

Both statements are true, but they say different things, so "the rate went up by 1%" is confusing. When you subtract one percent from another, say "percentage points".

Percentage Calculator

Type two numbers in any row. The answer and the calculation appear as you type.

Percents in Everyday Life

Percents compare parts of different wholes on the same scale, out of 100. That's why you meet them everywhere:

• shopping: discounts and sales tax;
• money: interest on savings and loans, tips, price increases;
• school: test scores and grades;
• news: election results, surveys, population growth;
• science: the strength of a solution, the humidity of the air, the nutrients on a food label.

Example 1. In class A, 18 of the 24 students passed a test. In class B, 21 of the 30 students passed. Which class did better?

Solution: More students passed in class B, but class B is also bigger. Compare the percents:

Class A: ${\frac{18}{24}\times 100\%}={75\%}$

Class B: ${\frac{21}{30}\times 100\%}={70\%}$

Class A did better.

Example 2. Tom runs a small grocery store. In his first month he bought goods for $\$650$ and sold them for $\$800$. In the second month he bought goods for $\$800$ and sold them for $\$1{,}200$. Did his business do better in the second month?

Solution: His profit grew from ${{800-650}={\$150}}$ to ${{1{,}200-800}={\$400}}$, but he also spent more. To see how well his money worked, compare each month's profit with the money he spent:

${\frac{150}{650}\times 100\%}\approx{23.1\%}$

${\frac{400}{800}\times 100\%}={50\%}$

In the first month every $\$100$ Tom spent brought him about $\$23$ of profit, and in the second month it brought $\$50$. So his business didn't just grow, it also became more profitable.

Common Mistakes

Wrong	Right
✗ ${8\%=0.8}$	✓ ${8\%=0.08}$
✗ ${0.5\%=0.5}$	✓ ${0.5\%=0.005}$
✗ From 80 to 120 is an increase of ${{\frac{40}{120}}\approx{33\%}}$	✓ Divide by the old value: ${\frac{40}{80}}={50\%}$
✗ +10% and then −10% is no change	✓ ${1.1\times 0.9}={0.99}$, a 1% decrease
✗ 20% off and then 10% off is 30% off	✓ ${0.8\times 0.9}={0.72}$, 28% off
✗ $\$64$ after 20% off, so the original price was ${64+0.2\times 64}={\$76.80}$	✓ ${64\div 0.8}={\$80}$
✗ 50 is 25% more than 40, so 40 is 25% less than 50	✓ 40 is ${\frac{10}{50}}={20\%}$ less than 50
✗ From 4% to 5% is an increase of 1%	✓ An increase of 1 percentage point, which is 25%

A good habit is to check that the answer makes sense: a price after a discount is lower than before, and 30% of a number is less than half of it.

Solved Problems

Problem 1. The price of a laptop was cut from $\$900$ to $\$765$. By what percent was the price reduced?

Solution: The price went down by ${{900-765}={\$135}}$. Compare that with the old price:

${\frac{135}{900}\times 100\%}={15\%}$

Problem 2. 200 g of salt water contains 15% salt. How much salt is in it? What percent of salt will there be after you add 100 g of water?

Solution: There is ${{0.15\times 200}=30}$ g of salt. After you add the water, there are ${{200+100}=300}$ g of salt water, but still 30 g of salt:

${\frac{30}{300}\times 100\%}={10\%}$

Problem 3. 2,400 people voted in an election with two candidates. Candidate A got 55% of the votes. How many more votes did A get than B?

Solution: B got ${{100\%-55\%}={45\%}}$ of the votes. The difference is ${{55\%-45\%}={10\%}}$ of the votes:

${0.1\times 2{,}400}=240$

Check: A got ${{0.55\times 2{,}400}=1{,}320}$ votes and B got ${{0.45\times 2{,}400}=1{,}080}$ votes, and ${{1{,}320-1{,}080}=240}$.

Problem 4. After a 25% discount, a shirt costs $\$18$. What was its price before the discount?

Solution: $\$18$ is ${{100\%-25\%}={75\%}}$ of the old price:

${18\div 0.75}={\$24}$

Problem 5. A town of 50,000 people shrank by 10% in one year and grew by 10% the next year. How many people live there now?

${50{,}000\times 0.9\times 1.1}={49{,}500}$

That is 500 people fewer than at the start, a 1% decrease.

Problem 6. A store raised all its prices by 25%. Later it put everything on sale at 20% off. Are the sale prices higher or lower than the prices before the increase?

Solution: ${{1.25\times 0.8}=1}$, so the sale prices are exactly the same as the old prices.

Problem 7. Unemployment fell from 8% to 6%. Describe the change in percentage points and in percent.

Solution: Unemployment fell by ${{8-6}=2}$ percentage points. As a percent change,

${\frac{6-8}{8}\times 100\%}={-25\%}$

so unemployment fell by 25%.

Practice

Try each problem, then open it to check your answer.

1. Write 45% as a decimal and as a fraction in lowest terms.

${45\%}={0.45}={\frac{45}{100}}={\frac{9}{20}}$

2. Write $\frac{7}{8}$ as a percent.

${7\div 8}={0.875}={87.5\%}$

3. Find 35% of 240.

${0.35\times 240}=84$

4. 27 is what percent of 36?

${\frac{27}{36}\times 100\%}={75\%}$

5. 18 is 45% of what number?

${18\div 0.45}=40$

6. A pair of shoes costs $\$60$ and is 15% off. What is the sale price?

${60\times 0.85}={\$51}$

7. The value of a car fell from $\$25{,}000$ to $\$21{,}000$. By what percent did it fall?

${\frac{25{,}000-21{,}000}{25{,}000}\times 100\%}={\frac{4{,}000}{25{,}000}\times 100\%}={16\%}$

8. After a 10% price increase, a bus ticket costs $\$2.75$. What did it cost before?

${2.75\div 1.1}={\$2.50}$

9. A price goes up by 30% and then down by 30%. What is the total change?

${{1.3\times 0.7}=0.91}$, so the price ends up 9% lower than at the start.

10. Which is more: 40% of 70 or 70% of 40?

They are equal: ${{0.4\times 70}=28}$ and ${{0.7\times 40}=28}$.