A princess is as old as the prince will be when the princess is twice as old as the prince was when the princess’ age was half the sum of their present age. What are their ages?
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.