Chapter 11.2, Problem 39WE

To determine

To find: The product of the numbers after converting into fractional form

Answer to Problem 39WE

The product of the two numbers is $\frac{2035}{1792}$

Explanation of Solution

Given information:

The given rational number is:

  $\begin{array}{l}{n}_1=0.\overline{407}\\ n_2=0.2\overline{7}\end{array}$

Converting the decimal expansion of a number in fractional form:

  $\begin{array}{c}{n}_1=0.\overline{407}\\ =\frac{407}{999}\end{array}$

  $n_2=0.2\overline{7}$

Since there are 1 digit in the repeating block, multiply the rational number by 10.

  $\begin{array}{l}10n_2=0.2\overline{7}\times 10\\ 10n_2=2.\overline{7}\\ 10n_2=2+0.\overline{7}\\ 10n_2=2+\frac{7}{9}\end{array}$

Simplify further,

  $\begin{array}{l}10n_2=\frac{25}{9}\\ n_2=\frac{25}{90}\end{array}$

Calculating the product of two numbers:

  $\begin{array}{c}N=n_1\times n_2\\ =\frac{407}{999}\times \frac{25}{90}\\ =\frac{2035}{1792}\end{array}$

Thus, the product of the two numbers is $\frac{2035}{1792}$