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Percents

Percent means “per hundred”: a fraction with 100 on the bottom.

Percents are everywhere: sales, tips, test scores, phone batteries. 25% means 25 out of every 100, the same as 25/100 or 0.25.

This page shows percents on a hundred grid, how to switch between fractions, decimals and percents, how to find a percent of a number, the whole or the percent, and how discounts, tax, tips and percent change work. Then try the practice problems and the Percent Shop game.

Jump to practicePlay Percent Shop
A 10 by 10 grid of 100 squares with 25 squares shaded, labeled 25 out of 100 = 25% 25 of 100 = 25%25 per hundred

What percent means

Percent comes from the Latin per centum, “by the hundred.” A percent is a ratio compared to 100. On a hundred grid, every square is 1%, and the whole grid is 100%.

A 10 by 10 hundred grid with 37 of the 100 squares shaded, showing 37% means 37 out of 100 37% = 37 out of 10037 shaded63 not shaded37 + 63 = 100
37% of the grid is shaded: 37 out of 100 squares.

Example 1: Reading a hundred grid

In a class survey, 37 out of 100 students walk to school. What percent walk? What percent do not?

  1. 37 out of 100 = 37/100 = 37%.
  2. The whole is 100%, so 100% − 37% = 63% do not walk.

Fractions, decimals and percents

These are three ways to write the same amount. To go from a fraction to a percent, find an equivalent fraction over 100 (see Fractions) or divide top by bottom to get a decimal. To go from a decimal to a percent, multiply by 100 (move the point two places right). To go back, divide by 100.

A hundred grid with 75 of its 100 squares shaded, showing 3/4 = 75/100 = 0.75 = 75% 3/4 = 75/100 = 0.75 = 75%fraction: 3/4per hundred: 75/100decimal: 0.75percent: 75%
Three quarters of the grid is 75 squares: 3/4, 75/100, 0.75 and 75% all name the same amount.

Example 2: Converting three ways

Write 3/4 as a decimal and a percent. Then write 0.4 as a fraction and a percent.

  1. 3/4 = (3 × 25)/(4 × 25) = 75/100 = 0.75 = 75%.
  2. 0.4 = 4/10 = 40/100 = 40%, and as a fraction in lowest terms 4/10 = 2/5.

Percent of a number

“Of” means multiply. To find p% of a number, write p% as a decimal or fraction and multiply. A double number line shows the same thing: percents on top, amounts below, lined up.

A double number line. The top line is marked 0%, 25%, 50%, 75% and 100%; the bottom line is marked 0, 20, 40, 60 and 80, so 25% of 80 is 20 25% of 80 = ?0%025%2050%4075%60100%80Split 100% into 4 equal jumps: 80 ÷ 4 = 20
100% lines up with the whole, 80. Four equal jumps of 25% are four jumps of 20.

Example 3: Percent of a number

What is 25% of 80? What is 15% of 60?

  1. 25% = 0.25 = 1/4, and 1/4 of 80 = 80 ÷ 4 = 20.
  2. 15% = 0.15, and 0.15 × 60 = 9. (Or: 10% of 60 is 6, 5% is half of that, 3, and 6 + 3 = 9.)

Finding the whole or the percent

Every percent problem has three numbers: the part, the whole and the percent, linked by part = percent × whole. If you know two, you can find the third. You can write a proportion, part/whole = p/100 (like in Ratios and Rates), or an equation and solve it with inverse operations (like in Simple Equations).

Example 4: Finding the percent

Kai got 18 out of 24 questions right. What percent is that?

  1. Part ÷ whole: 18 ÷ 24 = 0.75.
  2. 0.75 × 100 = 75%. (Proportion: 18/24 = p/100, so p = 75.)

Example 5: Finding the whole

12 students is 30% of a class. How many students are in the class?

  1. Write an equation: 0.30 × w = 12.
  2. Undo × 0.30 by dividing both sides by 0.30: w = 12 ÷ 0.3 = 40 students.
  3. Check: 30% of 40 = 0.3 × 40 = 12. ✓

Increase, decrease, discounts, tax and tips

A discount takes a percent off; tax and tips add a percent on. Find the percent of the price, then subtract or add. Shortcut: 25% off means you pay 75%, so multiply by 0.75; adding 8% tax means you pay 108%, so multiply by 1.08.

Percent change = change ÷ original amount, written as a percent. Always divide by the original (starting) amount.

A percent bar for a $60 jacket split into four 25% parts of $15 each. One part is the $15 discount, and the other three parts, 75%, are the $45 sale price $60 jacket, 25% off$1525%$1525%$1525%$1525%$15 offyou pay 75%: $45
Each quarter of the bar is 25% of $60, or $15. Take one quarter off and you pay the other 75%.

Example 6: Discount and tax

A $60 jacket is 25% off. Then 8% sales tax is added to the sale price. What is the total?

  1. Discount: 0.25 × 60 = $15, so the sale price is 60 − 15 = $45 (or 0.75 × 60 = $45).
  2. Tax: 0.08 × 45 = $3.60.
  3. Total: 45 + 3.60 = $48.60 (or 1.08 × 45 = 48.60).

Example 7: Percent increase

A plant grew from 20 cm to 26 cm. What is the percent increase? What would a 15% tip be on a $40 meal?

  1. Change = 26 − 20 = 6 cm. Change ÷ original = 6 ÷ 20 = 0.30 = 30% increase.
  2. Tip: 10% of $40 is $4, 5% is $2, so 15% = $6, and the total is $46.

Percents over 100% and under 1%

More than 100% means more than the whole: 125% = 1.25, one whole and a quarter more. Less than 1% is a small part of one square: 0.5% = 0.005, half of one hundredth.

Left: one full hundred grid and a second grid with 25 squares shaded, showing 125%. Right: a hundred grid with only half of one square shaded, showing 0.5% 125% = 1.25two grids: 100 + 25 squares0.5%half a square
125% fills one whole grid and a quarter of another. 0.5% is only half of one square.

Example 8: Big and small percents

What is 150% of 40? What is 0.5% of 600?

  1. 150% = 1.5, and 1.5 × 40 = 60 (more than 40, because 150% is more than the whole).
  2. 0.5% = 0.5/100 = 0.005, and 0.005 × 600 = 3. (1% of 600 is 6, so half a percent is 3.)

Common mistakes

  • Percents of different wholes don’t cancel. Lose 50% of $100 and you have $50. Gain 50% of $50 and you have $75, not $100.
  • Dividing by the new amount in percent change. From $50 to $75 is a 25 ÷ 50 = 50% increase, not 25 ÷ 75.
  • Forgetting to move the decimal: 5% is 0.05, not 0.5.
  • Adding percents of different things: 20% off and then another 10% off is 28% off in total, not 30%, because the 10% comes off the lower price.
Three bars. $100; then $50 after a 50% loss; then $75 after a 50% gain on $50. A dashed outline shows the $75 bar is shorter than the original $100 Lose 50%, then gain 50%$100$50 (−50%)$75 (+50%)Not back to $100: the second 50% is of a smaller whole
The 50% gain is half of $50, which is smaller than the 50% loss, which was half of $100.

How do we know percent of a number is multiplying?

p% means p/100, a fraction. Taking a fraction of a number means multiplying: 1/4 of 80 is 80 split into 4 equal parts, which is 80 × 1/4. So p% of n = (p/100) × n. Every percent problem is part = (p/100) × whole, and finding the whole or the percent is just solving that equation for a different unknown.

The same reason explains the 50% mistake: the first 50% is (1/2) × 100 and the second is (1/2) × 50. After both, you have 100 × 0.5 × 1.5 = 100 × 0.75, so you end with 75% of what you started with.

History: hundredths and the % sign

Long before decimals, people in ancient Rome often worked with fractions in hundredths. For example, the emperor Augustus charged a tax of 1/100 on goods sold at auction, called the centesima rerum venalium (“the hundredth of things sold”). Working with hundredths like this is the same as working with percents.

In the Middle Ages, merchants and arithmetic books, especially in Italy, wrote rates “for a hundred,” in Italian per cento, often shortened to forms like “p cento” or “pc” with a small circle. Over many years of copying, the abbreviation turned into a little circle, a line and another circle; one book from 1684 shows a version of it. The slanted form we type today, %, came later still.

Did you know?

No known text from before 1425 uses a special percent symbol. Historians trace the sign to abbreviations of per cento in Italian arithmetic manuscripts from the 1400s.

Practice problems

Try each one, then open it to compare.

1. Write 7/20 as a decimal and a percent.

7/20 = 35/100 = 0.35 = 35%.

2. What is 30% of 150?

0.3 × 150 = 45. (10% is 15, and 3 × 15 = 45.)

3. 14 is what percent of 56?

14 ÷ 56 = 0.25 = 25%.

4. 18 is 60% of what number?

0.6 × w = 18, so w = 18 ÷ 0.6 = 30. Check: 0.6 × 30 = 18. ✓

5. A $80 pair of shoes is 35% off. What is the sale price?

You pay 65%: 0.65 × 80 = $52. (Discount 0.35 × 80 = $28, and 80 − 28 = 52.)

6. A town’s population went from 2,000 to 2,500. What is the percent increase?

Change 500 ÷ original 2,000 = 0.25 = 25% increase.

7. A $50 meal has 6% tax and then a 20% tip on the $50. What is the total?

Tax 0.06 × 50 = $3; tip 0.2 × 50 = $10. Total 50 + 3 + 10 = $63.

8. A $100 game goes up 10%, then down 10%. What is the new price?

Up: 1.1 × 100 = $110. Down: 0.9 × 110 = $99, not $100, because the 10% decrease is of $110.

Game: Percent Shop

Ten percent problems. Every answer is a whole number. Type it, like 36, and press Enter. One point per correct answer. Your best score is saved only on this device. About 4 minutes.

Talk About It

  • A store says “up to 70% off.” What does that tell you, and what doesn’t it?
  • Is 20% off and then 20% off the same as 40% off? Explain with a $100 example.
  • Can a test score be 110%? Can a price rise 110%? What is the difference?
  • Which is easier in your head, 15% of 80 or 80% of 15? Why are they equal?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami. Names and prices in the examples are made up for practice.