Henry Ernest Dudeney/Puzzles and Curious Problems/2 - A Legacy Puzzle/Solution

by : $2$

 * A Legacy Puzzle
 * A man left legacies to his three sons and to a hospital, amounting in all to $\pounds 1,320$.
 * If he had left the hospital legacy also to his first son, that son would have received as much as the other two sons together.
 * If he had left it to his second son, that son would have received twice as much as the other two sons together.
 * If he had left the hospital legacy to his third son, he would have received then thrice as much as the first son and second son together.


 * Find the amount of each legacy.

Solution

 * The legacy to the first son was $\pounds 55$,
 * to the second son $\pounds 275$,
 * to the third son $\pounds 385$,
 * and to the hospital $\pounds 605$,
 * making $\pounds 1320$ in all.

Proof
Let $a$, $b$ and $c$ pounds be the legacies of the first, second and third sons respectively.

Let $h$ be the legacy of the hospital.

We have:

We set up this system of linear simultaneous equations in matrix form as:


 * $\begin {pmatrix}

1 & 1 &  1 &  1 \\ 1 & -1 & -1 &  1 \\ -2 &  1 & -2 &  1 \\ -3 & -3 &  1 &  1 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ h \end {pmatrix} = \begin {pmatrix} 1320 \\ 0 \\ 0 \\ 0 \end {pmatrix}$

It remains to solve this matrix equation.

In reduced echelon form, this gives:

from which the legacies can be read off directly.