Question

The difference of the two natural numbers is $5$ and the difference of their reciprocals is $\frac{1}{10}$. Find the numbers.

Answer 1

Let $x$ and $y$ be two natural numbers such that $x>y$

then Difference of the two natural numbers $=x-y=5$

Difference of their reciprocals $=\frac{1}{y}-\frac{1}{x}=\frac{1}{10}$ since $\frac{1}{y}>\frac{1}{x}$

$\Rightarrow \frac{x-y}{xy}=\frac{1}{10}$

$\Rightarrow \frac{5}{xy}=\frac{1}{10}$

$\Rightarrow xy=\frac{5}{\frac{1}{10}}=50$

$\Rightarrow x=\frac{50}{y}$

Using $x-y=5\Rightarrow \frac{50}{y}-y=5$

$\Rightarrow 50-y^2-5y=0$

$\Rightarrow y^2+5y-50=0$

$\Rightarrow y^2+10y-5y-50=0$

$\Rightarrow y(y+10)-5(y+10)=0$

$\Rightarrow (y+10)(y-5)=0$

$\Rightarrow y=-10,5$

$\therefore y=5$ since $y=-10$ is not a natural number.

$\Rightarrow x-y=5$

$\Rightarrow x-5=5$ or $x=10$

$\therefore$ the required natural numbers are $x=10,y=5$