Chapter 11.4, Problem 37PPS

To determine

To write the repeating decimal $2.\overline{18}$ as a fraction.

Answer to Problem 37PPS

  $\frac{24}{11}$

Explanation of Solution

Given:

  $2.\overline{18}$

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 equations 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 $2.\overline{18}$ as a fraction.

First note that there are two repeating digits(1, and 8), now take:

  $x=2.18181818$

Now multiply above equation by 100(since there are two repeating digit), as

  $\begin{array}{c}100\times x=100\times 2.18181818\\ 100x=218.18181818\end{array}$

Note that since 1, and 8 are repeating digits so 1, and 8 can be repeated after decimal any times.

Now, subtract $x=2.18181818$ from $100x=218.18181818$ and find value of x , as

  $\begin{array}{c}100x=218.18181818\\ -x=2.18181818\\ 99x=216.000000\\ x=\frac{216}{99}\\ \Rightarrow x=\frac{24}{11}\end{array}$

Thus, the given repeating decimal can be written into fraction equals to $\frac{24}{11}$ .