Henry Ernest Dudeney/Puzzles and Curious Problems/48 - Robinson's Age/Solution

by : $48$

 * Robinson's Age
 * Robinson said.
 * My brother is two years older than I,
 * my sister is four years older than he,
 * my mother was $20$ when I was born,
 * and I was told yesterday that the average age of the four of is is $39$ years.


 * What was Robinson's age?

Solution

 * $32$

Proof
The information can be represented as a system of linear simultaneous equations in matrix form as:


 * $\begin {pmatrix}

-1 & 1 &  0 &  0 \\ 0 & -1 & -1 &  0 \\ -1 &  0 &  0 &  1 \\ 1 &  1 &  1 &  1 \\ \end {pmatrix} \begin {pmatrix} R \\ B \\ S \\ M \end {pmatrix} = \begin {pmatrix} 2 \\ 4 \\ 20 \\ 4 \times 39 \end {pmatrix}$

where $R$, $B$, $S$ and $M$ are apparent.

It remains to solve this matrix equation.

Conversion to echelon form proceeds as follows:

Hence Robinson's mother is $52$, making Robinson $32$.

His brother is therefore $34$ and his sister $38$.