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# Repeating Decimals

## Identify repeating decimals by dividing the numerator of a fraction by the denominator.

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Repeating Decimals

Jason is helping his older sister babysit for one week during the summer. Jason's sister tells him that she will give him one third of the money she makes babysitting. Jason knows that his sister is expecting to receive 385 for the week of babysitting, so he knows he will receive 3853\begin{align*}\frac{385}{3}\end{align*} dollars. How can Jason write the amount of money he will receive as a decimal? In this concept, you will learn how to write fractions and mixed numbers as repeating decimals. ### Writing Fractions and Mixed Numbers as Repeating Decimals Every fraction is equivalent to some decimal. Some fractions are equivalent to terminating decimals. A terminating decimal is a decimal that ends and has a finite number of digits. Here are some examples of terminating decimals. • 0.5 • 0.75 • 0.1111 • 0.23985 Some fractions are equivalent to repeating decimals. A repeating decimal is a decimal that has digits that repeat over and over forever. You can indicate the digits that repeat by putting a line above these digits. Here are some examples of repeating decimals. • 0.33333333\begin{align*}0.33333333 \ldots\end{align*} or 0.3¯\begin{align*}0.\bar{3}\end{align*} • 0.123123123\begin{align*}0.123123123 \ldots\end{align*} or 0.123¯¯¯¯¯¯¯¯\begin{align*}0.\overline{123}\end{align*} • 0.166666666\begin{align*}0.166666666 \ldots\end{align*} or 0.16¯\begin{align*}0.1 \bar{6}\end{align*} The process for writing a fraction as a repeating decimal is the same as the process for writing a fraction as a terminating decimal. Here are the steps for writing a fraction as a decimal. 1. Rewrite your fraction as a division problem. 2. Divide using long division. Add a decimal point and zeros to the dividend as needed. Once you find a repeating pattern, stop dividing. Put a line above the repeating digits in your answer. Here is an example. Convert 56\begin{align*}\frac{5}{6}\end{align*} to a decimal. First, write 56\begin{align*}\frac{5}{6}\end{align*} as a division problem. 56\begin{align*}\frac{5}{6}\end{align*} is the same as 5÷6\begin{align*}5 \div 6\end{align*}. Next, use long division to divide. Watch for a repeating pattern. 6)5.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 0.8333 48 20 1820 18 20\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ 0.8333 \ldots}{6 \overline{ ) {5.0000 \;}}}\\ && \ \underline{- 48 \; \; \; \; \; \; \; }\\ && \quad \ \ 20 \\ && \underline{\ \ - 18 \; \; \; \; \; }\\ && \qquad 20 \\ && \underline{\quad \ - 18 \; \; \; }\\ && \qquad \ \ 20 \end{array}\end{align*} Notice that you have found a repeating pattern with the division. 6 always goes into 20 three times with a remainder of 2. The digit of 3 will repeat over and over at the end of the decimal. Write a line above the first digit of 3 to indicate that it repeats. 0.83¯\begin{align*}0.8 \bar{3}\end{align*} The answer is 56=0.83¯\begin{align*}\frac{5}{6} = 0.8 \bar{3}\end{align*}. You can also convert mixed numbers to repeating decimals. Here are the steps for writing mixed numbers as decimals. 1. Convert the fractional part of the mixed number to a decimal using long division. Add a decimal point and zeros to the dividend as needed. Once you find a repeating pattern, stop dividing. 2. Add the whole number part of the mixed number to the result from step 1. Put a line above the repeating digits in your answer. Here is an example. Convert 223\begin{align*}2 \frac{2}{3}\end{align*} to a decimal. First, set aside the 2. 23\begin{align*}\frac{2}{3}\end{align*} is the same as 2÷3\begin{align*}2 \div 3\end{align*}. Convert the 23\begin{align*}\frac{2}{3}\end{align*} to a decimal using long division. 3)2.000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 0.666 18 20 18 20\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ 0.666 \ldots}{3 \overline{ ) {2.000 \;}}}\\ && \ \underline{- 18 \; \; \; \; \; \; \; }\\ && \quad \ \ \ 20 \\ && \ \underline{\ \ - 18 \; \; \; \; \; }\\ && \qquad \ \ 20 \\ \end{array}\end{align*} The digit of 6 will repeat over and over at the end of the decimal. Write a line above the first digit of 6 to indicate that it repeats. 23=0.6¯\begin{align*}\frac{2}{3} = 0. \bar{6}\end{align*} Next, add the 2 from the original mixed number. 2.6¯\begin{align*}2.\bar{6}\end{align*} The answer is 223=2.6¯\begin{align*}2 \frac{2}{3} = 2. \bar{6}\end{align*}. ### Examples #### Example 1 Earlier, you were given a problem about Jason, who is helping his sister to babysit this summer. Jason's sister will give him one third of what she makes babysitting. Since Jason's sister is expecting to receive385 for the week of babysitting, Jason knows he will receive 3853\begin{align*}\frac{385}{3}\end{align*}. Jason wants to write the amount of money he will receive as a decimal.

First, Jason should write 3853\begin{align*}\frac{385}{3}\end{align*} as a mixed number.

3853=12813\begin{align*}\frac{385}{3} = 128 \frac{1}{3}\end{align*}

Next, Jason should set aside the 128. 13\begin{align*}\frac{1}{3}\end{align*} is the same as 1÷3\begin{align*}1 \div 3\end{align*}. He can convert the 13\begin{align*}\frac{1}{3}\end{align*} to a decimal using long division.

3)1.000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯     0.333  9   10 9  10\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ \ 0.333 \ldots}{3 \overline{ ) {1.000 \;}}}\\ && \underline{\ \ - 9 \; \; \; }\\ && \quad \ \ \ 10 \\ && \underline{\quad \ - 9 \; \; \;}\\ && \qquad \ \ 10 \\ \end{array}\end{align*}

The digit of 3 will repeat over and over at the end of the decimal. Jason could write a line above the first digit of 3 to indicate that it repeats.

13=0.3¯\begin{align*}\frac{1}{3} = 0. \bar{3}\end{align*}

Next, Jason can add the 128 from the original mixed number.

128.3¯\begin{align*}128.\bar{3}\end{align*}

Because 128.3¯\begin{align*}128.\bar{3}\end{align*} is an amount of money, Jason can round the number to the hundredths place.

The answer is Jason can expect to receive \$128.33 for helping his sister babysit.

#### Example 2

Write 216\begin{align*}2 \frac{1}{6}\end{align*} as a decimal.

First, set aside the 2. 16\begin{align*}\frac{1}{6}\end{align*} is the same as 1÷6\begin{align*}1 \div 6\end{align*}. Convert the 16\begin{align*}\frac{1}{6}\end{align*} to a decimal using long division.

6)1.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯     0.1666  6   40 36     40  3640\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ \ 0.1666 \ldots}{6 \overline{ ) {1.0000 \;}}}\\ && \underline{\ \ - 6 \; \; \; }\\ && \quad \ \ \ 40 \\ && \underline{\quad \ - 36 \; \; \;}\\ && \quad \ \ \ \ \ 40 \\ && \underline{\quad \ \ - 36 \; \; \; }\\ && \qquad \quad 40 \\ \end{array}\end{align*}

The digit of 6 will repeat over and over at the end of the decimal. Write a line above the first digit of 6 to indicate that it repeats.

16=0.16¯\begin{align*}\frac{1}{6} = 0.1 \bar{6}\end{align*}

Next, add the 2 from the original mixed number.

2.16¯\begin{align*}2.1 \bar{6}\end{align*}

The answer is 216=2.16¯\begin{align*}2 \frac{1}{6} = 2.1 \bar{6}\end{align*}

#### Example 3

Convert 16\begin{align*}\frac{1}{6}\end{align*} to a decimal.

First, write 16\begin{align*}\frac{1}{6}\end{align*} as a division problem.

16\begin{align*}\frac{1}{6}\end{align*} is the same as 1÷6\begin{align*}1 \div 6\end{align*}.

Next, use long division to divide. Watch for a repeating pattern.

6)1.0000¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯     0.1666    6   40    36      40       36        40\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ \ 0.1666 \ldots}{6 \overline{ ) {1.0000 \;}}}\\ && \underline{\ \ \ \ - 6 \; \; \; }\\ && \quad \ \ \ 40 \\ && \underline{\ \ \ \ - 36 \; \; \;}\\ && \quad \ \ \ \ \ \ 40 \\ && \underline{\ \ \ \ \ \ \ - 36 \; \; \; }\\ && \quad \ \ \ \ \ \ \ \ 40 \\ \end{array}\end{align*}

The digit of 6 will repeat over and over at the end of the decimal. Write a line above the first digit of 6 to indicate that it repeats.

\begin{align*}0.1 \bar{6}\end{align*}

The answer is \begin{align*}\frac{1}{6} = 0.1 \bar{6}\end{align*}

#### Example 4

Convert \begin{align*}4 \frac{4}{6}\end{align*} to a decimal.

First, set aside the 4. \begin{align*}\frac{4}{6}\end{align*} is the same as \begin{align*}4 \div 6\end{align*}. Convert the \begin{align*}\frac{4}{6}\end{align*} to a decimal using long division.

\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ \ 0.666 \ldots}{6 \overline{ ) {4.000 \;}}}\\ && \ \underline{ - 36 \; \; \; }\\ && \quad \ \ 40 \\ && \underline{\ \ - 36 \; \; \;}\\ && \quad \ \ \ \ 40 \\ \end{array}\end{align*}

The digit of 6 will repeat over and over at the end of the decimal. Write a line above the first digit of 6 to indicate that it repeats.

\begin{align*}\frac{4}{6} = 0.6 \bar{6}\end{align*}

Next, add the 4 from the original mixed number.

\begin{align*}4.\bar{6}\end{align*}

The answer is \begin{align*}4 \frac{4}{6} = 4. \bar{6}\end{align*}.

#### Example 5

Convert \begin{align*}\frac{4}{9}\end{align*} to a decimal.

First, write \begin{align*}\frac{4}{9}\end{align*} as a division problem.

\begin{align*}\frac{4}{9}\end{align*} is the same as \begin{align*}4 \div 9\end{align*}.

Next, use long division to divide. Watch for a repeating pattern.

\begin{align*}\begin{array}{rcl} && \overset{\ \ \ \ \ 0.444 \ldots}{9 \overline{ ) {4.000 \;}}}\\ && \ \underline{ - 36 \; \; \; }\\ && \quad \ \ 40 \\ && \underline{\ \ - 36 \; \; \;}\\ && \quad \ \ \ \ 40 \\ \end{array}\end{align*}

The digit of 4 will repeat over and over at the end of the decimal. Write a line above the first digit of 4 to indicate that it repeats.

\begin{align*}0. \bar{4}\end{align*}

The answer is \begin{align*}\frac{4}{9} = 0. \bar{4}\end{align*}.

### Review

Write each fraction or mixed number as a decimal.

1. \begin{align*}\frac{2}{3}\end{align*}
2. \begin{align*}\frac{5}{6}\end{align*}
3. \begin{align*}5 \frac{1}{3}\end{align*}
4. \begin{align*}\frac{1}{3}\end{align*}
5. \begin{align*}\frac{3}{4}\end{align*}
6. \begin{align*}\frac{1}{6}\end{align*}
7. \begin{align*}\frac{1}{8}\end{align*}
8. \begin{align*}\frac{3}{7}\end{align*}
9. \begin{align*}\frac{2}{6}\end{align*}
10. \begin{align*}4 \frac{2}{3}\end{align*}
11. \begin{align*}7 \frac{1}{3}\end{align*}
12. \begin{align*}6 \frac{5}{6}\end{align*}
13. \begin{align*}8 \frac{1}{2}\end{align*}
14. \begin{align*}9 \frac{2}{3}\end{align*}
15. \begin{align*}11 \frac{1}{5}\end{align*}

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### Vocabulary Language: English

TermDefinition
Decimal In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).
fraction A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
Mixed Number A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.

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