Repeating Decimals

Repeating decimals appear regularly on the GED Mathematical Reasoning test, especially when you convert fractions to decimals or classify rational numbers. A repeating decimal is any decimal whose digits repeat in a predictable pattern without end. Recognizing and working with repeating decimals is a key skill for rational number fluency.

What Is a Repeating Decimal?

A repeating decimal (also called a recurring decimal) is a decimal in which one or more digits repeat in a cycle forever. It is written with a bar (vinculum) placed over the repeating digit(s). For example:

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• \(\color{blue}{\frac{1}{3} = 0.333\ldots = 0.&\#773;3}\)
• \(\color{blue}{\frac{2}{9} = 0.222\ldots = 0.&\#773;2}\)
• \(\color{blue}{\frac{1}{7} = 0.142857142857\ldots = 0.142857}\)

Repeating decimals are rational numbers — they can always be written as fractions.

Terminating vs. Repeating Decimals

Terminating decimals

A decimal that ends (has a finite number of digits) after the decimal point. These come from fractions whose denominator (in lowest terms) has only factors of 2 and/or 5.

• \(\color{blue}{\frac{3}{4} = 0.75}\) (terminates)
• \(\color{blue}{\frac{7}{20} = 0.35}\) (terminates)

Repeating decimals

A decimal that repeats forever. These come from fractions whose denominator (in lowest terms) has a prime factor other than 2 or 5.

• \(\color{blue}{\frac{1}{3} = 0.&\#773;3}\) (3 is neither 2 nor 5)
• \(\color{blue}{\frac{5}{6} = 0.8&\#773;3}\) (\(\color{blue}{6 = 2 \times 3}\); the factor 3 causes the repeat)
• \(\color{blue}{\frac{7}{11} = 0.63}\) (11 is neither 2 nor 5)

Step-by-Step Summary

1. Convert the fraction to a decimal by dividing the numerator by the denominator.
2. If the long division terminates, it is a terminating decimal.
3. If a remainder repeats, the decimal repeats; write the repeating block under a bar.
4. To convert a repeating decimal back to a fraction: let \(\color{blue}{x = 0.&\#773;d}\), multiply both sides to shift the decimal, subtract, and solve.

Watch: Converting a Fraction to a Repeating Decimal (Video Lesson)

Khan Academy walks through converting \(\color{blue}{\frac{19}{27}}\) to a repeating decimal with long division:

Repeating Decimals – Worked Examples

Example 1: Convert \(\color{blue}{\frac{1}{3}}\) to a decimal. Is it repeating?

Divide: \(\color{blue}{1.000 \div 3}\). Remainders cycle: \(\color{blue}{1, 1, 1, \ldots}\)
\(\color{blue}{\frac{1}{3} = 0.333\ldots = 0.&\#773;3}\). Yes, it is repeating.

Example 2: Convert \(\color{blue}{\frac{2}{9}}\) to a decimal.

Divide: \(\color{blue}{2.000 \div 9}\). Remainders cycle: \(\color{blue}{2, 2, 2, \ldots}\)
\(\color{blue}{\frac{2}{9} = 0.222\ldots = 0.&\#773;2}\).

Example 3: Convert \(\color{blue}{\frac{5}{6}}\) to a decimal.

Divide: \(\color{blue}{5 \div 6 = 0.8333\ldots}\) The digit 8 does not repeat, but 3 does.
\(\color{blue}{\frac{5}{6} = 0.8&\#773;3}\).

Example 4: Convert the repeating decimal \(\color{blue}{0.&\#773;3}\) back to a fraction.

Let \(\color{blue}{x = 0.333\ldots}\)
Then \(\color{blue}{10x = 3.333\ldots}\)
Subtract: \(\color{blue}{10x – x = 3.333\ldots – 0.333\ldots = 3}\)
\(\color{blue}{9x = 3}\), so \(\color{blue}{x = \frac{3}{9} = \frac{1}{3}}\).

More Practice: Converting Repeating Decimals to Fractions (Video)

Khan Academy shows the algebraic method for converting any repeating decimal to a fraction:

Exercises for Repeating Decimals

1. Convert \(\color{blue}{\frac{1}{6}}\) to a decimal. Is it repeating?
2. Convert \(\color{blue}{\frac{7}{9}}\) to a decimal.
3. Convert \(\color{blue}{\frac{5}{11}}\) to a decimal.
4. Is \(\color{blue}{\frac{3}{8}}\) a terminating or repeating decimal? Justify your answer.
5. Convert \(\color{blue}{0.&\#773;6}\) to a fraction in lowest terms.
6. Convert \(\color{blue}{0.27}\) (i.e., \(\color{blue}{0.272727\ldots}\)) to a fraction.

Answers

1. \(\color{blue}{\frac{1}{6} = 0.1&\#773;6}\); yes, it is repeating (\(\color{blue}{6 = 2 \times 3}\); factor 3 causes the repeat).
2. \(\color{blue}{\frac{7}{9} = 0.&\#773;7}\)
3. \(\color{blue}{\frac{5}{11} = 0.45}\)
4. Terminating: \(\color{blue}{\frac{3}{8} = 0.375}\). The denominator \(\color{blue}{8 = 2^{3}}\) has only the factor 2.
5. Let \(\color{blue}{x = 0.666\ldots}\); \(\color{blue}{10x = 6.666\ldots}\); \(\color{blue}{9x = 6}\); \(\color{blue}{x = \frac{6}{9} = \frac{2}{3}}\).
6. Let \(\color{blue}{x = 0.272727\ldots}\); \(\color{blue}{100x = 27.272727\ldots}\); \(\color{blue}{99x = 27}\); \(\color{blue}{x = \frac{27}{99} = \frac{3}{11}}\).

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Frequently Asked Questions

Are repeating decimals rational numbers?

Yes. Every repeating decimal can be expressed as a fraction of two integers, which is exactly the definition of a rational number. The algebraic method (multiply, subtract, solve) always produces a fraction.

How can I tell if a fraction will produce a repeating decimal?

Simplify the fraction first. If the denominator’s prime factors are only 2s and 5s, the decimal terminates. If the denominator has any other prime factor (3, 7, 11, …), the decimal repeats.

How do repeating decimals appear on the GED test?

GED questions may ask you to convert a fraction to a decimal and identify whether it repeats, classify a decimal as rational or irrational, or convert a repeating decimal back to a fraction.

Related Topics

• Convert Between Fractions and Decimals
• Using a Venn Diagram to Classify Rational Numbers
• Simplifying Fractions
• Convert Between Decimals and Mixed Numbers
• Order of Decimals Mixed Numbers and Fractions