Five plus the reciprocal

Statement

Five, added to the reciprocal of a number, equals $9$ divided by that number. What is the number?

Solution

Unknown. Let $x$ be the number, with $x\neq 0$.

Equation. The reciprocal is $\dfrac{1}{x}$ and ”$9$ divided by the number” is $\dfrac{9}{x}$: $5 + \frac{1}{x} = \frac{9}{x}.$ Move the terms with $x$ to the same side: $5 = \frac{9}{x} - \frac{1}{x} = \frac{8}{x} \implies 5x = 8 \implies x = \frac{8}{5}.$ Check: $5 + \dfrac{1}{8/5} = 5 + \dfrac{5}{8} = \dfrac{45}{8}$ and $\dfrac{9}{8/5} = \dfrac{45}{8}$. Correct. $\boxed{x = \frac{8}{5} = 1.6}$