# Solution to Practice Problem #6

This is a hard problem, and the solution below is advanced. I need to teach it properly only after building correct foundations.

Question: A father is three times as old as his son was at the time when the man was twice as old as his son will be in two years. What are the current ages of the father and son if the sum of their ages now is 55 years.

\begin{array}{c} \eqalign{ & F = 3\left( {S – t} \right) \cr & \cr & F – t = 2\left( {S + 2} \right) \cr & \cr & F + S = 55 \cr & \cr & F – 3S + 3t = 0 \cr & F – 2S – t = 4 \cr & F + S + 0 = 55 \cr & \cr & rref\left[ {\matrix{ 1 & { – 3} & 3 & 0 \cr 1 & { – 2} & { – 1} & 4 \cr 1 & 1 & 0 & {55} \cr } } \right] \cr & \cr & = \left[ {\matrix{ 1 & 0 & 0 & {39} \cr 0 & 1 & 0 & {16} \cr 0 & 0 & 1 & 3 \cr } } \right] \cr} \end{array}

The father is 39 years old, the son is 16 years old, and the time span is 3 years.