Question

The difference of two natural numbers is 5 and the difference of their reciprocal is 110. find the numbers

Answer 1

$x - y = 5 \\ \frac{1}{y} - \frac{1}{x} = \frac{1}{10} \\ \frac{x - y}{xy} = \frac{1}{10} \\ \frac{5}{xy} = \frac{1}{10} \\ xy = 50 \\ x(x - 5) = 50\\ {x}^{2} - 5x = 50\\ {x }^{2} - 5x - 50 = 0 \\ {x}^{2} - 10x + 5x - 50 = 0 \\ (x - 10)(x + 5) = 0 \\ x = 10 \\ y = 5$