Question

The sum of a natural number and its reciprocal is $\frac{10}{3}$. Find the number.

Answer 1

Let the number be x. Then, its reciprocal will be $\frac{1}{x}$.
By the given condition,
$$\begin{gathered} x+\frac{1}{x}=\frac{10}{3} \\ \Rightarrow \frac{x^2+1}{x}=\frac{10}{3} \\ \Rightarrow 3x^2+3=10x \\ \Rightarrow 3x^2-10x+3=0 \end{gathered}$$

On splitting the middle term –10x as –x – 9x, we get:
3x² – x – 9x + 3 = 0
$\Rightarrow$x(3x – 1) – 3(3x – 1) = 0
$\Rightarrow$(3x – 1)(x – 3) = 0
$\Rightarrow$3x – 1 = 0 or x – 3 = 0
$\Rightarrow$x = $\frac{1}{3}$ or x = 3
Since x is a natural number, it cannot be fraction.
Therefore, x = 3.
Thus, the required number is 3.