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

It is given that the difference between two natural numbers = 5.

Let one number be x. So, the other number = x + 5

And, the difference of their reciprocals is $\dfrac{1}{10}$.

$\Rightarrow \dfrac{1}{x} - \dfrac{1}{x + 5} = \dfrac{1}{10}\\[1em] \Rightarrow \dfrac{x + 5}{x(x + 5)} - \dfrac{x}{x(x + 5)} = \dfrac{1}{10}\\[1em] \Rightarrow \dfrac{x + 5 - x}{x(x + 5)} = \dfrac{1}{10}\\[1em] \Rightarrow \dfrac{5}{x(x + 5)} = \dfrac{1}{10}\\[1em] \Rightarrow 5 \times 10 = x(x + 5)\\[1em] \Rightarrow 50 = x^2 + 5x\\[1em] \Rightarrow x^2 + 5x - 50 = 0\\[1em] \Rightarrow x^2 + 10x - 5x - 50 = 0\\[1em] \Rightarrow x(x + 10) - 5(x + 10) = 0\\[1em] \Rightarrow (x + 10)(x - 5) = 0\\[1em] \Rightarrow (x + 10) = 0 \text{ or }(x - 5) = 0\\[1em] \Rightarrow x = -10 \text{ or }x = 5$

It is given that numbers are natural numbers. So, number cannot be -10.

∴ x = 5

Other number = 5 + 5 = 10

Hence, the two numbers are 5 and 10.