Question

If the sum of the fraction and its reciprocal is $2\frac{16}{21}$, find the fraction.

Answer 1

Let $x$ be the required fraction.

Therefore,

$x+\frac{1}{x}=2\frac{16}{21}$

$x+\frac{1}{x}=\frac{58}{21}$

$\Rightarrow 21x^2+21=58x$

$\Rightarrow 21x^2-58x+21=0$

$\Rightarrow 21x^2-9x-49x+21=0$

$\Rightarrow (7x-3)(3x-7)=0$

$\Rightarrow x=\frac{7}{3}$ or $x=\frac{3}{7}$

Hence the required fraction can be $\frac{3}{7}$ or $\frac{7}{3}$.