The princess is 40 years old and the prince is 30 years old (there are multiple solutions).

Let current age of prince = X
Current age of princess = Y
Years in future = P
Years in past = Q

Equation 1: Y = X + P
Equation 2: Y + P = 2(X – Q)
Equation 3: Y – Q = 1/2(Y + X)

We have 3 equations, with 4 unknowns. This implies that there are multiple solutions. Let’s set everything equal to 0 first (and multiply equation 3 by two to make it easier).

Y – X – P = 0
Y + P + 2Q – 2X = 0
Y – 2Q – X = 0

We’re looking for the age of the prince and age of the princess (X and Y), so we want P and Q to equal zero. Instead of using a matrix, I took advantage of the convenient situation (look at the values of P and Q in these 3 equations – what happens when you add them?) and added all the equations together.

Therefore: 3Y – 4X = 0

Either: (1) y = 4/3X OR (2) X = 3/4Y

Which in English means: The prince is 3/4 the age of the princess.