Henry Ernest Dudeney/Modern Puzzles/93 - Sum Equals Product/Solution

Modern Puzzles by Henry Ernest Dudeney: $93$

There are two numbers whose sum equals their product, that is, $2$ and $2$.

What other numbers have that property?

Any $x$ and $y$ such that $y = \dfrac x {x - 1}$ fits the bill.

$\ds x + y$

$=$

$\ds x y$

$\ds \leadsto \ \ $

$\ds y \paren {x - 1}$

$=$

$\ds x$

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds \dfrac x {x - 1}$

and we note that:

$\ds x + \dfrac x {x - 1}$

$=$

$\ds \dfrac {x \paren {x - 1} + x} {x - 1}$

$\ds $

$=$

$\ds \dfrac {x^2} {x - 1}$

$\ds $

$=$

$\ds x \dfrac x {x - 1}$

$\blacksquare$