problem: Solve the following equation for $x$: \[ \ \frac{2}{3} = \frac{4}{x-5}.\]
solution: Multiplying both sides by $x-5$ and by 3 gives $2(x-5) = 4(3)$.  Expanding the left side gives $2x-10 = 12$.  Adding 10 to both sides gives $2x = 22$ and dividing by 2 gives $x = \boxed{11}$.