Sum of two natural numbers is 8 and the difference of their reciprocal is $\dfrac{2}{15}.$ Find the numbers.

Let the first number be x

Since, the sum of two numbers is 8, so other number is 8 - x.

Given, the difference of reciprocal of numbers = $\dfrac{2}{15}$

$\Rightarrow \dfrac{1}{x} - \dfrac{1}{8 - x} = \dfrac{2}{15} \\[0.5em] \Rightarrow \dfrac{8 - x - x}{x(8 - x)} = \dfrac{2}{15} \text{ (On taking L.C.M.) } \\[0.5em] \Rightarrow 15(8 - 2x) = 2x(8 - x) \text{ (On cross multiplication ) } \\[0.5em] \Rightarrow 120 - 30x = 16x - 2x^2 \\[0.5em] \Rightarrow 120 - 30x - 16x + 2x^2 = 0 \\[0.5em] \Rightarrow 2x^2 - 46x + 120 = 0 \\[0.5em] \Rightarrow 2(x^2 - 23x + 60) = 0 \\[0.5em] \Rightarrow x^2 - 20x - 3x + 60 = 0 \\[0.5em] \Rightarrow x(x - 20) - 3(x - 20) = 0 \\[0.5em] \Rightarrow (x - 3)(x - 20) = 0 \text{ (Factorising left side) } \\[0.5em] \Rightarrow x - 3 = 0 \text{ (or) } x - 20 = 0 \text{ (Zero-product rule) } \\[0.5em] \Rightarrow x = 3 \text{ or } x = 20.$

If x = 20 , 8 - x = -12 , Since both are natural numbers hence x ≠ 20.

∴ x = 3 , 8 - x = 5

Hence, the required natural numbers are 3 , 5.