Fractions, decimals and percents

3/8, 0.375 and 37.5% are the same amount written three ways.

Shade three of eight and read what comes out

Before any rule, use the bar below. It starts with 8 equal parts.

Shade 3 of 8 parts and the reading under the bar is 0.375 and 37.5%. Switch to 5 parts and shade 2: 0.4 and 40%. Switch to 3 parts and shade 1: the bar rounds the repeating decimal to 0.333 and the percent to 33.3%.

One shaded area, three numbers. They are the same amount: 3/8 of the bar, 0.375 of it, 37.5% of it. The rest of the page shows how to get from any one of them to the other two when there is no bar to shade.

45 squares out of 100

OpenStax defines a percent as a ratio whose denominator is 100. The sign % is a reminder of the 100 hidden below, so 45% is 45 out of every 100, or 45/100. Shade 45 squares of a 10 by 10 grid and you have drawn 45%, 45/100 and 0.45 in one picture.

Why 8% is 0.08 and not 0.8

Because a percent is hundredths, changing between percent and decimal is a move of two places. To turn a percent into a decimal, drop the sign and move the point two places left: 45% = 0.45, 8% = 0.08, 7.5% = 0.075. To turn a decimal into a percent, move the point two places right and add the sign: 0.6 = 60%, 0.125 = 12.5%, 1.25 = 125%. Percents above 100 simply mean more than the whole, and percents below 1 are tiny: 0.4% is 0.004.

A point moved only one place, or moved the wrong way, is easy to catch. A percent under 100 must become a decimal under 1. So 8% cannot be 0.8. That is 80%.

3/8 is 3 divided by 8

A fraction is a division, so a fraction becomes a decimal when you divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375, which is 37.5%. When the denominator already divides into 10, 100 or 1,000, you can skip the division: 7/20 = 35/100 = 0.35, because 20 × 5 = 100.

Some divisions never end. 1/3 = 0.333..., with the 3 repeating forever, and 2/3 = 0.666.... A repeating decimal is often written with a bar over the repeating part, or rounded: 1/3 is about 33.3%. Fractions in simplest form whose denominators have only 2s and 5s as prime factors, such as 1/4, 3/8 and 7/20, always end; others, such as 1/3, 1/6 and 1/7, repeat.

From 12.5% back to 1/8

Write the percent over 100 and simplify. 45% = 45/100 = 9/20. 150% = 150/100 = 3/2. For a percent with a decimal, clear the decimal first: 12.5% = 12.5/100 = 125/1,000 = 1/8.

Four short problems from everyday life

Example 1. A test has 50 questions and you get 42 right. 42/50 = 84/100, so the score is 84%. Doubling both numbers turned the denominator into 100 without any division. Example 2. A fuel gauge shows 7/8 of a tank. 7 ÷ 8 = 0.875, so the tank is 87.5% full. Example 3. A savings account pays 0.6% a year. As a decimal that is 0.006, so 500 dollars earns 0.006 × 500 = 3 dollars in a year. Example 4. Is 2/7 more or less than 30%? 2 ÷ 7 = 0.2857..., about 28.6%, so it is a little less. These numbers are made up for practice.