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
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $48$. -- Robinson's Age
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $45$. Robinson's Age