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

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Puzzles and Curious Problems by Henry Ernest Dudeney: $48$

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:

\(\ds \) \(\) \(\ds \paren {\begin {array} {cccc{{|}}c}

-1 & 1 & 0 & 0 & 2 \\ 

0 & -1 &  1 &  0 &   4 \\

-1 & 0 & 0 & 1 & 20 \\ 

1 &  1 &  1 &  1 & 156 \\ \end {array} }\)
\(\ds \) \(\leadsto\) \(\ds \paren {\begin {array} {cccc{{|}}c} 
1 & -1 &  0 &  0 &  -2 \\
0 &  1 & -1 &  0 &  -4 \\
1 &  0 &  0 & -1 & -20 \\
1 &  1 &  1 &  1 & 156 \\ \end {array} }\)
$r_1 \to -r_1$, $r_2 \to -r_2$, $r_3 \to -r_3$
\(\ds \) \(\leadsto\) \(\ds \paren {\begin {array} {cccc{{|}}c} 
1 & -1 &  0 &  0 &  -2 \\
0 &  1 & -1 &  0 &  -4 \\
0 &  1 &  0 & -1 & -18 \\
0 &  2 &  1 &  1 & 158 \\ \end {array} }\)
$r_3 \to r_3 - r_1$, $r_4 \to r_4 - r_1$
\(\ds \) \(\leadsto\) \(\ds \paren {\begin {array} {cccc{{|}}c} 
1 & -1 &  0 &  0 &  -2 \\
0 &  1 & -1 &  0 &  -4 \\
0 &  0 &  1 & -1 & -14 \\
0 &  0 &  3 &  1 & 166 \\ \end {array} }\)
$r_4 \to r_4 - 2 r_2$
\(\ds \) \(\leadsto\) \(\ds \paren {\begin {array} {cccc{{|}}c} 
1 & -1 &  0 &  0 &  -2 \\
0 &  1 & -1 &  0 &  -4 \\
0 &  0 &  1 & -1 & -14 \\
0 &  0 &  0 &  4 & 208 \\ \end {array} }\)
$r_4 \to r_4 - 3 r_3$
\(\ds \) \(\leadsto\) \(\ds \paren {\begin {array} {cccc{{|}}c} 
1 & -1 &  0 &  0 &  -2 \\
0 &  1 & -1 &  0 &  -4 \\
0 &  0 &  1 & -1 & -14 \\
0 &  0 &  0 &  1 &  52 \\ \end {array} }\)
$r_4 \to r_4 / 4$

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

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

$\blacksquare$


Sources

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