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

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