Henry Ernest Dudeney/Modern Puzzles/22 - Mrs. Wilson's Family

by : $22$

 * Mrs. Wilson's Family
 * Mrs. Wilson had three children, Edgar, James and John.
 * Their combined ages were half of hers.
 * Five years later, during which time Ethel was born, Mrs. Wilson's age equalled the total of all her children's ages.
 * Ten years more have passed, Daisy appearing during that interval.
 * At the latter event Edgar was as old as John and Ethel together.
 * The combined ages of all the children are now double Mrs. Wilson's age, which is, in fact, only equal to that of Edgar and James together.
 * Edgar's age also equals that of the two daughters.


 * Can you find all their ages?

Solution

 * Edgar is $21$
 * James and John are both $18$ (they are twins)
 * Ethel is $12$
 * Daisy is $9$
 * Mrs. Wilson herself is $39$.

Proof
The wording is not completely clear, but it is assumed that "now" is the same time as "ten years more have passed".

So, let us define the following variables:


 * Let $a$, $b$, $c$, $d$ and $e$ be the current ages of Edgar, James, John, Daisy and Ethel.
 * Let $M$ be the current age of Mrs. Wilson.
 * Let $x$ be the number of years before now that Daisy appeared.

We can set up a set of equations encoding the statements in the question as follows:

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


 * $\begin {pmatrix}

2 & 2 & 2 &  0 &  0 & -1 \\ 1 & 1 &  1 &  0 &  1 & -1 \\ 1 & 0 & -1 &  1 & -1 &  0 \\ 1 & 1 &  1 &  1 &  1 & -2 \\ 1 & 1 &  0 &  0 &  0 & -1 \\ 1 & 0 &  0 & -1 & -1 &  0 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \\ M \end {pmatrix} = \begin {pmatrix} 75 \\ 30 \\ 0 \\ 0 \\ 0 \\ 0  \end {pmatrix}$

It remains to solve this matrix equation.

Rearranging into a more convenient order for conversion into reduced echelon form:


 * $\begin {pmatrix}

1 & 0 & -1 & 1 & -1 &  0 \\ 1 & 1 &  1 &  1 &  1 & -2 \\ 1 & 1 &  0 &  0 &  0 & -1 \\ 1 & 0 &  0 & -1 & -1 &  0 \\ 1 & 1 &  1 &  0 &  1 & -1 \\ 2 & 2 &  2 &  0 &  0 & -1 \\ \end {pmatrix} \begin {pmatrix} a \\ b \\ c \\ d \\ e \\ M \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 30 \\ 75 \\ \end {pmatrix}$

In reduced echelon form, this gives:

from which the ages can be read off directly.