Chapter 11.4, Problem 15CYU

To determine

To write the repeating decimal $0.\overline{642}$ as a fraction.

Answer to Problem 15CYU

  $\frac{214}{333}$

Explanation of Solution

Given:

  $0.\overline{642}$

Concept Used:

To write a repeating decimal as a fraction:

1. Write x equals to the repeating decimal.
2. Place the repeating digit to the left of the decimal point.
3. The number of repeating digits in both equation taken on right side of the decimal point should be equal.
4. Solve the above two equations formed to find the value of x .
5. The value of x is required fraction for the given repeating decimal.

Calculation:

In order to write the repeating decimal $0.\overline{642}$ as a fraction.

First note that there are three repeating digits(6, 4, and 2) thus take:

  $x=0.652652652$

Now place the repeating digit to the left of the decimal point for that multiply above by 1000(since there are three repeating digit), as

  $\begin{array}{c}1000\times x=1000\times 0.642642642\\ 1000x=642.642642642\end{array}$

Note that the number of repeating digits in both equations taken on right side of the decimal point should be equal.

Now, subtract $x=0.642642642$ from $1000x=642.642642642$ and find value of x , as

  $\begin{array}{c}1000x=642.642642642\\ -x=0.642642642\\ 999x=642.000000\\ x=\frac{642}{999}\\ \Rightarrow x=\frac{214}{333}\end{array}$

Thus, the given repeating decimal can be written into fraction equals to $\frac{214}{333}$ .