3.16重复小数

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  重复小数

  

  [Figure 1]

  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 $\begin{array}{r}\frac{385}{3}\end{array}$ 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.

  • $\begin{array}{r}0.33333333\dots \end{array}$ or $\begin{array}{r}0.\overline{3}\end{array}$
  • $\begin{array}{r}0.123123123\dots \end{array}$ or $\begin{array}{r}0.\overline{123}\end{array}$
  • $\begin{array}{r}0.166666666\dots \end{array}$ or $\begin{array}{r}0.1\overline{6}\end{array}$

  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 $\begin{array}{r}\frac{5}{6}\end{array}$ to a decimal.

  First, write $\begin{array}{r}\frac{5}{6}\end{array}$ as a division problem.

  $\begin{array}{r}\frac{5}{6}\end{array}$ is the same as $\begin{array}{r}5÷6\end{array}$.

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

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{0.8333\dots }{6\overline{)5.0000}}\\ & & \underset{\_}{-48}\\ & & 20\\ & & \underset{\_}{-18}\\ & & 20\\ & & \underset{\_}{-18}\\ & & 20\end{array}\end{array}$$

  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.

  $$\begin{array}{r}0.8\overline{3}\end{array}$$

  The answer is $\begin{array}{r}\frac{5}{6}=0.8\overline{3}\end{array}$.

  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 $\begin{array}{r}2\frac{2}{3}\end{array}$ to a decimal.

  First, set aside the 2. $\begin{array}{r}\frac{2}{3}\end{array}$ is the same as $\begin{array}{r}2÷3\end{array}$. Convert the $\begin{array}{r}\frac{2}{3}\end{array}$ to a decimal using long division.

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{0.666\dots }{3\overline{)2.000}}\\ & & \underset{\_}{-18}\\ & & 20\\ & & \underset{\_}{-18}\\ & & 20\end{array}\end{array}$$

  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.

  $$\begin{array}{r}\frac{2}{3}=0.\overline{6}\end{array}$$

  Next, add the 2 from the original mixed number.

  $$\begin{array}{r}2.\overline{6}\end{array}$$

  The answer is $\begin{array}{r}2\frac{2}{3}=2.\overline{6}\end{array}$.

  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 receive  $385 for the week of babysitting, Jason knows he will receive $\begin{array}{r}\frac{385}{3}\end{array}$. Jason wants to write the amount of money he will receive as a decimal.

  First, Jason should write $\begin{array}{r}\frac{385}{3}\end{array}$ as a mixed number.

  $$\begin{array}{r}\frac{385}{3}=128\frac{1}{3}\end{array}$$

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

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{0.333\dots }{3\overline{)1.000}}\\ & & \underset{\_}{-9}\\ & & 10\\ & & \underset{\_}{-9}\\ & & 10\end{array}\end{array}$$

  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.

  $$\begin{array}{r}\frac{1}{3}=0.\overline{3}\end{array}$$

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

  $$\begin{array}{r}128.\overline{3}\end{array}$$

  Because $\begin{array}{r}128.\overline{3}\end{array}$ 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 $\begin{array}{r}2\frac{1}{6}\end{array}$ as a decimal.

  First, set aside the 2. $\begin{array}{r}\frac{1}{6}\end{array}$ is the same as $\begin{array}{r}1÷6\end{array}$. Convert the $\begin{array}{r}\frac{1}{6}\end{array}$ to a decimal using long division.

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{0.1666\dots }{6\overline{)1.0000}}\\ & & \underset{\_}{-6}\\ & & 40\\ & & \underset{\_}{-36}\\ & & 40\\ & & \underset{\_}{-36}\\ & & 40\end{array}\end{array}$$

  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.

  $$\begin{array}{r}\frac{1}{6}=0.1\overline{6}\end{array}$$

  Next, add the 2 from the original mixed number.

  $$\begin{array}{r}2.1\overline{6}\end{array}$$

  The answer is $\begin{array}{r}2\frac{1}{6}=2.1\overline{6}\end{array}$

  Example 3

  Convert $\begin{array}{r}\frac{1}{6}\end{array}$ to a decimal.

  First, write $\begin{array}{r}\frac{1}{6}\end{array}$ as a division problem.

  $\begin{array}{r}\frac{1}{6}\end{array}$ is the same as $\begin{array}{r}1÷6\end{array}$.

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

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{0.1666\dots }{6\overline{)1.0000}}\\ & & \underset{\_}{-6}\\ & & 40\\ & & \underset{\_}{-36}\\ & & 40\\ & & \underset{\_}{-36}\\ & & 40\end{array}\end{array}$$

  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.

  $$\begin{array}{r}0.1\overline{6}\end{array}$$

  The answer is $\begin{array}{r}\frac{1}{6}=0.1\overline{6}\end{array}$

  Example 4

  Convert $\begin{array}{r}4\frac{4}{6}\end{array}$ to a decimal.

  First, set aside the 4. $\begin{array}{r}\frac{4}{6}\end{array}$ is the same as $\begin{array}{r}4÷6\end{array}$. Convert the $\begin{array}{r}\frac{4}{6}\end{array}$ to a decimal using long division.

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{0.666\dots }{6\overline{)4.000}}\\ & & \underset{\_}{-36}\\ & & 40\\ & & \underset{\_}{-36}\\ & & 40\end{array}\end{array}$$

  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.

  $$\begin{array}{r}\frac{4}{6}=0.6\overline{6}\end{array}$$

  Next, add the 4 from the original mixed number.

  $$\begin{array}{r}4.\overline{6}\end{array}$$

  The answer is $\begin{array}{r}4\frac{4}{6}=4.\overline{6}\end{array}$.

  Example 5

  Convert $\begin{array}{r}\frac{4}{9}\end{array}$ to a decimal.

  First, write $\begin{array}{r}\frac{4}{9}\end{array}$ as a division problem.

  $\begin{array}{r}\frac{4}{9}\end{array}$ is the same as $\begin{array}{r}4÷9\end{array}$.

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

  $$\begin{array}{r}\begin{array}{rcl}& & \stackrel{0.444\dots }{9\overline{)4.000}}\\ & & \underset{\_}{-36}\\ & & 40\\ & & \underset{\_}{-36}\\ & & 40\end{array}\end{array}$$

  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.

  $$\begin{array}{r}0.\overline{4}\end{array}$$

  The answer is $\begin{array}{r}\frac{4}{9}=0.\overline{4}\end{array}$.

  Review

  Write each fraction or mixed number as a decimal.

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

  Review (Answers)

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