Henry Ernest Dudeney/Puzzles and Curious Problems/230 - In a Garden/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $230$

In a Garden
Consider a rectangular flower-bed.
If it were $2$ feet broader and $3$ feet longer, it would have been $64$ square feet larger;
if it were $3$ feet broader and $2$ feet longer, it would have been $68$ square feet larger.
What is its length and breadth?


Solution

The flower bed is $10$ feet broad and $14$ feet long.


Proof

Let $a$ and $b$ feet be the length and breadth respectively of the flower bed.

We have:

\(\text {(1)}: \quad\) \(\ds \paren {a + 3} \paren {b + 2}\) \(=\) \(\ds a b + 64\)
\(\text {(2)}: \quad\) \(\ds \paren {a + 2} \paren {b + 3}\) \(=\) \(\ds a b + 68\)
\(\text {(3)}: \quad\) \(\ds \leadsto \ \ \) \(\ds 2 a + 3 b\) \(=\) \(\ds 58\) simplifying $(1)$
\(\text {(4)}: \quad\) \(\ds 3 a + 2 b\) \(=\) \(\ds 62\) simplifying $(2)$
\(\ds \leadsto \ \ \) \(\ds \begin {pmatrix} 2 & 3 \\ 3 & 2 \end {pmatrix} \begin {pmatrix} a \\ b \end {pmatrix}\) \(=\) \(\ds \begin {pmatrix} 58 \\ 62 \end {pmatrix}\) expressing $(3)$ and $(4)$ in matrix form
\(\ds \leadsto \ \ \) \(\ds \begin {pmatrix} a \\ b \end {pmatrix}\) \(=\) \(\ds \dfrac 1 5 \begin {pmatrix} -2 & 3 \\ 3 & -2 \end {pmatrix} \begin {pmatrix} 58 \\ 62 \end {pmatrix}\) Inverse of Order 2 Square Matrix
\(\ds \) \(=\) \(\ds \begin {pmatrix} 14 \\ 10 \end {pmatrix}\) Definition of Matrix Product (Conventional)

Hence the result.

$\blacksquare$


Sources

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