Henry Ernest Dudeney/Puzzles and Curious Problems/24 - The Two Turkeys/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $24$

The Two Turkeys
"I sold those two turkeys," said Tozer.
"They weighed $20$ pounds together.
Mrs. Burkett paid $24 \shillings 8 \oldpence$ for the large one, and Mrs. Suggs paid $6 \shillings 10 \oldpence$ for the small one.
I made $2 \oldpence$ a pound more on the little one than on the other."
What did the big one weigh?


Solution

$16$ pounds.


Proof

Let $L$ and $S$ pounds be the weight of the large and small turkeys together.

Let all prices be considered in pence in order to reduce the need to handle fractions.

The price in pence per pound for the large turkey is:

$\dfrac {24 \times 12 + 8} L = \dfrac {296} L$

The price in pence per pound for the large turkey is:

$\dfrac {6 \times 12 + 10} S = \dfrac {82} S$

We have:

\(\text {(1)}: \quad\) \(\ds L + S\) \(=\) \(\ds 20\) "They weighed $20$ pounds together.
\(\text {(2)}: \quad\) \(\ds \dfrac {296} L + 2\) \(=\) \(\ds \dfrac {82} S\) I made $2 \oldpence$ a pound more on the little one than on the other."
\(\ds \leadsto \ \ \) \(\ds \dfrac {296} L + 2\) \(=\) \(\ds \dfrac {82} {20 - L}\)
\(\ds \leadsto \ \ \) \(\ds 296 \paren {L - 20} + 2 L \paren {L - 20} + 82 L\) \(=\) \(\ds 0\) multiplying through by $L \paren {L - 20}$ to clear fractions
\(\ds \leadsto \ \ \) \(\ds 2 L^2 + \paren {296 - 40 + 82} L - 296 \times 20\) \(=\) \(\ds 0\) collecting terms
\(\ds \leadsto \ \ \) \(\ds L^2 + 169 L - 2960\) \(=\) \(\ds 0\) simplifying
\(\ds \leadsto \ \ \) \(\ds L\) \(=\) \(\ds \dfrac {-169 \pm \sqrt {169^2 + 2 \times 2960} } 2\) Quadratic Formula
\(\ds \leadsto \ \ \) \(\ds L\) \(=\) \(\ds \dfrac {-169 \pm 201} 2\) simplifying
\(\ds \) \(=\) \(\ds 16 \text { or } -185\) simplifying

Only the positive root is required here.

Hence the result.

$\blacksquare$


Sources

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