Henry Ernest Dudeney/Puzzles and Curious Problems/119 - The Three Drovers/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $119$
- The Three Drovers
- Three drovers with varied flocks met on the highway.
- Said Jack to Jim: "If I give you six pigs for a horse then you will have twice as many animals in your drove as I will have in mine."
- Said Dan to Jack: "If I give you fourteen sheep for a horse, then you'll have three times as many animals as I have got."
- Said Jim to Dan: "But if I give you four cows for a horse, then you'll have six times as many animals as I."
- There were no deals; but can you tell me how many animals there were in the three droves?
Solution
Jack had $11$ animals, Jim had $7$, and Dan had $21$.
Proof
Let $a$, $b$ and $c$ be the numbers of animals in each of Jack's, Jim's and Dan's droves respectively.
We have:
\(\ds 2 \paren {a - 5}\) | \(=\) | \(\ds b + 5\) | Said Jack to Jim: "If I give you six pigs for a horse then you will have twice as many animals in your drove as I will have in mine." | |||||||||||
\(\ds 3 \paren {c - 13}\) | \(=\) | \(\ds a + 13\) | Said Dan to Jack: "If I give you fourteen sheep for a horse, then you'll have three times as many animals as I have got." | |||||||||||
\(\ds 6 \paren {b - 3}\) | \(=\) | \(\ds c + 3\) | Said Jim to Dan: "But if I give you four cows for a horse, then you'll have six times as many animals as I." | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 2 a - 15\) | \(=\) | \(\ds b\) | simplifying the above constraints | |||||||||
\(\text {(2)}: \quad\) | \(\ds 3 c - 52\) | \(=\) | \(\ds a\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds 6 b - 21\) | \(=\) | \(\ds c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 \paren {6 b - 21} - 52\) | \(=\) | \(\ds a\) | eliminating $c$ from $(2)$ and $(3)$ | ||||||||||
\(\text {(4)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 18 b - 115\) | \(=\) | \(\ds a\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 18 \paren {2 a - 15} - 115\) | \(=\) | \(\ds a\) | eliminating $b$ from $(1)$ and $(4)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 35 a\) | \(=\) | \(\ds 385\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds 11\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b\) | \(=\) | \(\ds 2 \times 11 - 15\) | substituting for $a$ in $(1)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 7\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds c\) | \(=\) | \(\ds 6 \times 7 - 21\) | substituting for $b$ in $(3)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 21\) | simplifying |
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $119$. -- The Three Drovers
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $193$. The Three Drovers