Chapter 11.2, Problem 40WE

To determine

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

Answer to Problem 40WE

The product of the two numbers is $\frac{1176}{2475}$

Explanation of Solution

Given information:

The given rational number is:

  $\begin{array}{l}{n}_1=0.3\overline{5}\\ n_2=1.3\overline{36}\end{array}$

Converting the decimal expansion of a number in fractional form:

  $n_1=0.3\overline{5}$

The given rational number can be converted into rational form as follows:

Multiply both sides by 10:

  $\begin{array}{l}{n}_1=0.3\overline{5}\\ 10n_1=3.\overline{5}\\ 10n_1=3+0.\overline{5}\\ 10n_1=3+\frac{5}{9}\end{array}$

Simplify further,

  $\begin{array}{l}10n_1=\frac{32}{9}\\ n_1=\frac{32}{90}\end{array}$

Converting the other decimal expansion in fractional form:

  $n_2=1.3\overline{36}$

Multiply both sides by 10:

  $\begin{array}{l}10n_2=13.\overline{36}\\ 10n_2=13+0.\overline{36}\\ 10n_2=13+\frac{36}{99}\\ n_2=\frac{1323}{990}\end{array}$

Now finding the product of the two numbers:

  $\begin{array}{c}N=n_1\times n_2\\ =\frac{32}{90}\times \frac{1323}{990}\\ =\frac{16}{45}\times \frac{147}{110}\\ =\frac{1176}{2475}\end{array}$

Thus, the product of the two numbers is $\frac{1176}{2475}$