Converting Fractions and Decimals

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Converting fractions and decimals works in both directions. To change a fraction to a decimal, divide the top by the bottom: \frac{3}{4} = 3 \div 4 = 0.75. To change a decimal to a fraction, write its digits over 10, 100 or 1,000 and simplify: 0.75 = \frac{75}{100} = \frac{3}{4}. Some fractions never end, like \frac{1}{3} = 0.333\ldots

Before this lesson: You should know how to simplify fractions and how to divide in operations with decimals.

Fractions and Decimals Name the Same Numbers

A fraction bar means divide. That is why every fraction can also be written as a decimal. They are two names for the same point on the number line.

Number line from 0 to 1. Above the line are the fractions 1/4, 1/2 and 3/4. Below the line are the matching decimals 0.25, 0.5 and 0.75. Each fraction and decimal pair names the same point.
Each fraction above the line names the same point as the decimal below it, such as 3/4 = 0.75.

Here are conversions that are worth knowing by heart:

FractionDecimalFractionDecimal
\frac{1}{2}0.5\frac{1}{5}0.2
\frac{1}{4}0.25\frac{1}{8}0.125
\frac{3}{4}0.75\frac{1}{10}0.1
\frac{1}{3}0.\overline{3}\frac{2}{3}0.\overline{6}

How to Convert Fractions and Decimals

Fraction to decimal

  1. Divide the numerator by the denominator. Add zeros after the decimal point as you need them.
  2. Stop when the remainder is 0. This is a terminating decimal.
  3. If the digits start to repeat forever, it is a repeating decimal. Draw a bar over the part that repeats.

A shortcut: if you can make the denominator 10, 100 or 1,000 by multiplying, do that. For example, \frac{3}{20} = \frac{15}{100} = 0.15.

Decimal to fraction

  1. Read the decimal using place value. The last digit’s place tells you the denominator.
  2. Write the digits over 10, 100 or 1,000.
  3. Simplify the fraction.

For example, 0.45 means 45 hundredths, so 0.45 = \frac{45}{100}.

Worked Examples

Example 1: A fraction that terminates

Write \frac{3}{8} as a decimal.

Step 1. Divide 3 by 8. Since 8 doesn’t go into 3, write 3 as 3.000.

Step 2. 8 goes into 30 three times with 6 left. 8 goes into 60 seven times with 4 left. 8 goes into 40 five times with 0 left.

Answer: \frac{3}{8} = 0.375. Check with the shortcut: \frac{3 \times 125}{8 \times 125} = \frac{375}{1000}.

Example 2: A fraction that repeats

Write \frac{2}{3} as a decimal.

Step 1. Divide 2 by 3. 3 goes into 20 six times with 2 left over.

Step 2. The remainder is 2 again, so the 6 repeats forever: 0.666…

Answer: \frac{2}{3} = 0.\overline{6}. Rounded to the nearest hundredth, it is about 0.67.

Example 3: A decimal to a fraction

Write 0.45 as a fraction in simplest form.

Step 1. The last digit is in the hundredths place, so the denominator is 100: 0.45 = \frac{45}{100}.

Step 2. The GCF of 45 and 100 is 5. Divide both by 5.

\frac{45}{100} = \frac{45 \div 5}{100 \div 5} = \frac{9}{20}

Answer: 0.45 = \frac{9}{20}.

Example 4: A decimal bigger than 1

Write 2.125 as a mixed number.

Step 1. Keep the whole number 2. The decimal part 0.125 has three decimal places, so it is \frac{125}{1000}.

Step 2. Simplify. The GCF of 125 and 1000 is 125, so \frac{125}{1000} = \frac{1}{8}.

Answer: 2.125 = 2\frac{1}{8}, which is the same as the improper fraction \frac{17}{8}.

Example 5: Word problem

In a survey, 13 out of 20 students chose pizza as their favorite lunch. Write this as a decimal.

Step 1. The fraction is \frac{13}{20}.

Step 2. Multiply the top and bottom by 5 to get a denominator of 100: \frac{13 \times 5}{20 \times 5} = \frac{65}{100}.

Answer: 0.65 of the students chose pizza.

Common Mistakes

MistakeWhat it looks likeHow to fix it
Writing the digits side by side\frac{3}{4} = 3.4The bar means divide: 3 \div 4 = 0.75.
Dividing the wrong way\frac{3}{4} = 4 \div 3Always divide the top by the bottom.
Reading the place value wrong0.08 = \frac{8}{10}The 8 is in the hundredths place, so 0.08 = \frac{8}{100} = \frac{2}{25}.
Treating a rounded decimal as exact\frac{2}{3} = 0.66Write 0.\overline{6}, or say “about 0.67” when you round.
Flipping a decimal into a unit fraction0.5 = \frac{1}{5}0.5 means 5 tenths: \frac{5}{10} = \frac{1}{2}.

Repeating Decimals and Comparing Numbers

Which fractions repeat

Simplify the fraction first. If the denominator has no prime factors except 2 and 5, the decimal ends. If it has any other prime factor, the decimal repeats. For example, \frac{7}{20} ends because 20 = 2^2 \times 5, but \frac{1}{6} repeats because 6 has a factor of 3. See prime factorization for help.

Comparing a fraction and a decimal

Change the fraction to a decimal, then compare. For example, \frac{5}{8} = 0.625, so it is greater than 0.6. Decimals and fractions also lead straight into percentages, since 0.65 is the same as 65%.

Practice Problems

For problems 1–4, write the fraction as a decimal. For problems 5–8, write the decimal as a fraction or mixed number in simplest form.

  1. \frac{1}{5}
  2. \frac{7}{20}
  3. \frac{5}{8}
  4. \frac{1}{6}
  5. 0.6
  6. 0.125
  7. 0.04
  8. 3.75
  9. Which is greater, \frac{5}{8} or 0.6?
  10. A recipe calls for 0.75 cup of milk. What fraction of a cup is that?
Show answers
  1. 1 ÷ 5 = 0.2
  2. \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35
  3. 5 ÷ 8 = 0.625
  4. 1 ÷ 6 = 0.1666…, so \frac{1}{6} = 0.1\overline{6}. Only the 6 repeats.
  5. \frac{6}{10} = \frac{3}{5}
  6. \frac{125}{1000} = \frac{1}{8}
  7. \frac{4}{100} = \frac{1}{25}
  8. 3\frac{75}{100} = 3\frac{3}{4}, or \frac{15}{4}
  9. \frac{5}{8} = 0.625, and 0.625 is greater than 0.600. So \frac{5}{8} is greater.
  10. 0.75 = \frac{75}{100} = \frac{3}{4} cup

Converting Fractions and Decimals FAQ

What is 1/3 as a decimal?

\frac{1}{3} = 0.333\ldots, which you can write as 0.\overline{3}. Rounded to the nearest hundredth, it is about 0.33.

How do you write a repeating decimal?

Put a bar over the digits that repeat. For example, 0.272727… is written 0.\overline{27}.

How do you know if a decimal will repeat?

Simplify the fraction. If its denominator has only 2s and 5s as prime factors, the decimal ends. Otherwise, it repeats.

Is 0.999… equal to 1?

Yes. Since \frac{1}{3} = 0.333\ldots, three times that is \frac{3}{3} = 0.999\ldots, and \frac{3}{3} = 1.

Summary

  • To change a fraction to a decimal, divide the numerator by the denominator.
  • A decimal either ends (terminates) or repeats. Use a bar to show repeating digits.
  • To change a decimal to a fraction, write the digits over 10, 100 or 1,000, then simplify.
  • The place of the last digit tells you the denominator.
  • Change both numbers to decimals when you want to compare them.

Previous lesson: Operations with Decimals

Next lesson: Nested Fractions

Back to the Pre-Algebra (Math Basics) Index

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