Percentages
Percentages are numbers out of 100. The symbol % means “per hundred,” so 25% means 25 out of 100, which is 0.25 or \frac{1}{4}. To find a percent of a number, change the percent to a decimal and multiply. For example, 25% of 80 is 0.25 \times 80 = 20.
Before this lesson: You should know how to convert fractions and decimals and how to solve proportions.
What Is a Percent?
“Percent” comes from words meaning “for each hundred.” Picture a grid of 100 squares. If 25 squares are shaded, then 25% of the grid is shaded.
So 100% means the whole thing, 50% means half and 0% means none of it.
Converting Percents, Decimals and Fractions
| To change | Do this | Example |
|---|---|---|
| Percent to decimal | Divide by 100. Move the decimal point 2 places left. | 35% = 0.35 |
| Decimal to percent | Multiply by 100. Move the decimal point 2 places right. | 0.6 = 60% |
| Percent to fraction | Write it over 100 and simplify. | 35\% = \frac{35}{100} = \frac{7}{20} |
| Fraction to percent | Change it to a decimal, then multiply by 100. | \frac{3}{8} = 0.375 = 37.5\% |
The Three Kinds of Percent Problems
Every basic percent problem has three parts: the part, the whole and the percent. One of them is missing.
\text{part} = \text{percent} \times \text{whole}| What is missing | What to do | Example |
|---|---|---|
| The part | Multiply the percent (as a decimal) by the whole. | 15% of 60 is 0.15 \times 60 = 9 |
| The percent | Divide the part by the whole, then multiply by 100. | 18 out of 24 is 18 \div 24 = 0.75 = 75\% |
| The whole | Divide the part by the percent (as a decimal). | 30 is 40% of 30 \div 0.40 = 75 |
You can also solve any of them with the percent proportion: \frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}.
Worked Examples
Example 1: Convert a percent and a fraction
Write 35% as a decimal and as a fraction. Then write \frac{3}{8} as a percent.
Step 1. Move the decimal point 2 places left: 35% = 0.35.
Step 2. Write it over 100 and simplify: \frac{35}{100} = \frac{7}{20}.
Step 3. For \frac{3}{8}, divide: 3 ÷ 8 = 0.375. Then move the decimal point 2 places right: 37.5%.
Answer: 35% = 0.35 = \frac{7}{20}, and \frac{3}{8} = 37.5%.
Example 2: Find the part
What is 15% of 60?
Step 1. Change 15% to a decimal: 0.15.
Step 2. Multiply by the whole: 0.15 \times 60 = 9.
Answer: 15% of 60 is 9.
Example 3: Find the percent
18 is what percent of 24?
Step 1. Divide the part by the whole: 18 ÷ 24 = 0.75.
Step 2. Change the decimal to a percent: 0.75 = 75%.
Answer: 18 is 75% of 24.
Example 4: Find the whole
30 is 40% of what number?
Step 1. Write the percent proportion: \frac{30}{x} = \frac{40}{100}.
Step 2. Cross multiply: 40x = 3000, so x = 75.
Answer: 30 is 40% of 75. Check: 0.40 \times 75 = 30.
Example 5: Word problem
A school asked 250 students about their favorite sport. 38% chose soccer. How many students chose soccer?
Step 1. You know the whole (250) and the percent (38%). You need the part.
Step 2. Multiply: 0.38 \times 250 = 95.
Answer: 95 students chose soccer.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Moving the decimal only one place | 5% = 0.5 | Move it 2 places: 5% = 0.05. |
| Moving the decimal the wrong way | 0.6 = 0.006% | To get a percent, multiply by 100: 0.6 = 60%. |
| Writing the digits of a fraction as a percent | \frac{3}{8} = 3.8\% | Divide first: 3 ÷ 8 = 0.375 = 37.5%. |
| Mixing up the part and the whole | “18 is what percent of 24?” solved as 24 ÷ 18 | The whole comes after “of.” Divide the part by the whole: 18 ÷ 24. |
| Thinking a percent can’t be over 100 | “150% is impossible” | More than 100% means more than the whole: 150% of 40 is 60. |
Percents Bigger Than 100 or Less Than 1
A percent over 100 means more than the whole. For example, 150% = 1.5, and 150% of 40 is 60.
A percent under 1 is a tiny part. For example, 0.5% = 0.005, which is half of 1%.
For mental math, find 10% by moving the decimal point 1 place left. 10% of 70 is 7, so 5% of 70 is half of that, 3.5. You’ll use percents to find increases and decreases in the next lesson, Percent Change.
Practice Problems
- Write 0.62 as a percent.
- Write 8% as a decimal.
- Write 45% as a fraction in simplest form.
- Write \frac{7}{20} as a percent.
- What is 20% of 45?
- What is 6% of 150?
- 12 is what percent of 48?
- 27 is 90% of what number?
- What is 150% of 40?
- A basketball player made 22 of 25 free throws. What percent did she make?
Show answers
- 0.62 × 100 = 62%
- 8 ÷ 100 = 0.08
- \frac{45}{100} = \frac{9}{20}
- 7 ÷ 20 = 0.35 = 35%
- 0.20 \times 45 = 9
- 0.06 \times 150 = 9
- 12 ÷ 48 = 0.25 = 25%
- 27 ÷ 0.9 = 30
- 1.5 \times 40 = 60
- 22 ÷ 25 = 0.88 = 88%
Percentages FAQ
How do you calculate a percentage of a number?
Change the percent to a decimal by dividing by 100, then multiply by the number. For example, 30% of 50 is 0.30 \times 50 = 15.
Can a percent be more than 100?
Yes. A percent over 100 means more than the whole amount. If a plant grows to 200% of its height, it is twice as tall.
What is 1/3 as a percent?
\frac{1}{3} = 0.333\ldots, so it is about 33.3% when rounded to the nearest tenth of a percent.
How do you find a percent in your head?
Find 10% first by moving the decimal point one place left. Then build from it: 20% is double 10%, and 5% is half of 10%.
Summary
- A percent is a number out of 100.
- Percent to decimal: move the decimal point 2 places left. Decimal to percent: move it 2 places right.
- To find the part, multiply the percent (as a decimal) by the whole.
- To find the percent, divide the part by the whole and multiply by 100.
- To find the whole, divide the part by the percent, or use the percent proportion.
Previous lesson: Unit Conversions
Next lesson: Percent Change
Back to the Pre-Algebra Index