Chapter 6.6, Problem 22WE

To determine

To write the given decimal as a common fraction in lowest terms.

Answer to Problem 22WE

  $1.\overline{101}=\frac{1100}{999}$

Explanation of Solution

Given information :

Given number is $1.\overline{101}$ .

The given number has repeated decimal digits. We know $1.\overline{101}=\mathrm{1.101101...}$

Now to convert this in fraction let us assume the number as $x$

Now $x=\mathrm{1.101101....}$ (Equation 1)

Multiply by 1000 on both sides we get,

  $1000x=\mathrm{1101.101101...}$ (Equation 2)

Now we subtract equation 1 from equation 2 we get,

  $\begin{array}{l}1000x-x=\left(\mathrm{1101.101101...}\right)-\left(\mathrm{1.101101...}\right)\\ 999x=1100\\ x=\frac{1100}{999}\end{array}$

Hence we get the value $1.\overline{101}=\frac{1100}{999}$