The sum of a number and its reciprocal is $\frac{65}{8}$ . Find the number.

Let the number be x.

According to the given question:

$\Rightarrow {\mathrm{x}+\frac{1}{\mathrm{x}}=\frac{65}{8}}_{ }$ 

$\Rightarrow \frac{{\mathrm{x}}^2+1}{\mathrm{x}}=\frac{65}{8}$

=> 8(x² + 1) = 65x

=> 8x² + 8 = 65x

=> 8x² - 65x + 8 = 0

=> 8x² - 64x - x + 8 = 0

=> 8x(x - 8) - 1(x - 8) = 0

=> (x - 8)(8x - 1) = 0 

$\Rightarrow {\mathrm{x}=8, \frac{1}{8}}_{ }$ 

Hence, The number is 8 or $\frac{1}{8}$.