Chapter 2.8, Problem 35WE

To determine

To find: the number whose sum with its reciprocal is $\frac{13}{6}$ .

Answer to Problem 35WE

  $\frac{3}{2}\text{ or }\frac{2}{3}$

Explanation of Solution

Given:

  $\text{Domain:}\left\{\frac{2}{3},\frac{3}{4},\frac{5}{4},\frac{3}{2}\right\}$

Calculation:

Let the number be x .

Then, its reciprocal is $\frac{1}{x}$ .

Given that their sum is $\frac{13}{6}$ . This can be translated as:

  $x+\frac{1}{x}=\frac{13}{6}$ ….. (1)

First substitute $x=\frac{2}{3}$ in equation (1),

  $\begin{array}{c}\frac{2}{3}+\frac{3}{2}=\frac{13}{6}\\ \frac{4+9}{6}=\frac{13}{6}\\ \frac{13}{6}=\frac{13}{6}\end{array}$

The above statement is true.

Now substitute $x=\frac{3}{4}$ in equation (1),

  $\begin{array}{c}\frac{3}{4}+\frac{4}{3}=\frac{13}{6}\\ \frac{9+16}{12}=\frac{13}{6}\\ \frac{25}{12}=\frac{13}{6}\end{array}$

The above statement is false.

Now substitute $x=\frac{5}{4}$ in equation (1),

  $\begin{array}{c}\frac{5}{4}+\frac{4}{5}=\frac{13}{6}\\ \frac{25+16}{20}=\frac{13}{6}\\ \frac{41}{20}=\frac{13}{6}\end{array}$

The above statement is false.

Now substitute $x=\frac{3}{2}$ in equation (1),

  $\begin{array}{c}\frac{3}{2}+\frac{2}{3}=\frac{13}{6}\\ \frac{9+4}{6}=\frac{13}{6}\\ \frac{13}{6}=\frac{13}{6}\end{array}$

The above statement is true.

Hence, the solution set is $\left\{\frac{2}{3},\frac{3}{2}\right\}$ .