Question

A number is the reciprocal of the other. If the arithmetic mean of the two numbers be $\frac{13}{12}$, then the numbers are

• A. $\frac{1}{4},\frac{4}{1}$
• B. $\frac{3}{4},\frac{4}{3}$
• C. $\frac{2}{5},\frac{5}{2}$
• D. $\frac{3}{2},\frac{2}{3}$

Answer 1

The correct option is D

$\frac{3}{2},\frac{2}{3}$

Explanation for the correct option:

Step 1. Finding the numbers:

Let, $x$ is be the number and other number reciprocal of $1$st.

$\therefore$the other number is $\frac{1}{x}$.

Arithmetic mean of the two numbers is $\frac{13}{12}$

Arithmetic mean of these two numbers $=\frac{(x+{\displaystyle \frac{1}{x}})}{2}$

$=\frac{x^2+1}{2x}$

Step 2. Put the value of AM in above equation, we get

$\begin{array}{rcl}AM& =& \frac{x^2+1}{2x}\\ & =& \frac{13}{12}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}12x^2+12& =& 26x\end{array}$

$\Rightarrow$$\begin{array}{rcl}6x^2-13x+6& =& 0\end{array}$

$\Rightarrow$ $\begin{array}{rcl}x& =& \frac{3}{2},\frac{2}{3}\end{array}$

Now,

If $x=\frac{3}{2}$,

$\Rightarrow$$\frac{1}{x}=\frac{2}{3}$

If $x=\frac{2}{3}$

$\Rightarrow$$\frac{1}{x}=\frac{3}{2}$

Hence, Option (D) is the correct answer.