Chapter 6.7, Problem 11MRE

To determine

To Find:

  $3.7\overline{82}$ is rational or irrational.

Answer to Problem 11MRE

Rational number

Explanation of Solution

Given :

The expression is :

  $3.7\overline{82}$

Concept Used:

Rational number has its decimal expansion either terminating or repeating.

Calculation:

Given:

  $\begin{array}{l}x=3.7\overline{82}\\ 100x=378.\overline{28}\\ 100x-x=378.2\overline{82}-3.7\overline{82}=374.5\\ 99x=374.5\\ x=\frac{374.5}{99}=\frac{3745}{990}=\frac{749}{198}\\ So,\\ 3.7\overline{82}=\frac{749}{198}\\ Hence,\\ 3.7\overline{82}\text{ is a rational number}\text{.}\end{array}$

  $3.7\overline{82}$ has its decimal expansion which is repeating.

So, it is rational.