My uncle and I

Statement

Four years ago my uncle was four times as old as I was. In two years he will be three times as old as I will be. How old am I now?

Solution

Unknowns. Let $x$ be my age today and $z$ my uncle’s age.

Equations. Four years ago we were $x-4$ and $z-4$; in two years we will be $x+2$ and $z+2$: $\begin{cases} z-4 = 4(x-4)\\ z+2 = 3(x+2) \end{cases}$ From the first: $z = 4x-16+4 = 4x-12$. Substituting into the second: $4x-12+2 = 3(x+2) \implies 4x-10 = 3x+6 \implies x = 16.$ Hence $z = 4\cdot 16 - 12 = 52$. Check: four years ago I was $12$ and my uncle $48=4\cdot 12$; in two years I will be $18$ and he $54=3\cdot 18$. Correct. $\boxed{\text{me } 16 \text{ years}, \quad \text{uncle } 52 \text{ years}}$