# The sum of a number and its reciprocal is

Question:

The sum of a number and its reciprocal is $2 \frac{1}{30}$. Find the number.

Solution:

Let the required number be x.

According to the given condition,

$x+\frac{1}{x}=2 \frac{1}{30}$

$\Rightarrow \frac{x^{2}+1}{x}=\frac{61}{30}$

$\Rightarrow 30 x^{2}+30=61 x$

$\Rightarrow 30 x^{2}-61 x+30=0$

$\Rightarrow 30 x^{2}-36 x-25 x+30=0$

$\Rightarrow 6 x(5 x-6)-5(5 x-6)=0$

$\Rightarrow(5 x-6)(6 x-5)=0$

$\Rightarrow 5 x-6=0$ or $6 x-5=0$

$\Rightarrow x=\frac{6}{5}$ or $x=\frac{5}{6}$

Hence, the required number is $\frac{5}{6}$ or $\frac{6}{5}$.