Liber Abaci/Problems/Lion, Leopard and Bear
Classic Problem
- A lion would take $4$ hours to eat $1$ sheep.
- A leopard would take $5$ hours.
- A bear would take $6$.
- If a single sheep were to be thrown to them, how many hours would it take to devour it?
Solution
The three together would consume the sheep in $1 \frac {23} {37}$ hours.
Proof
Using the Method of False Position:
For $4$ hours, in which the lion eats a sheep, put $\dfrac 1 4$.
For the $5$ hours the leopard takes, put $\dfrac 1 5$.
For the $6$ hours the bear takes, put $\dfrac 1 6$.
Because $\dfrac 1 6$, $\dfrac 1 5$ and $\dfrac 1 4$ are found exactly in $60$, suppose that in $60$ hours they devour the sheep.
Then consider how many sheep a lion can eat in $60$ hours;
- since in four hours it can devour one sheep, it can consume $15$ sheep in $60$ hours
and the leopard would eat $12$ as a fifth of $60$ is $12$.
Similarly the bear would eat $10$, as $\dfrac 1 6$ of $60$ is $10$.
Therefore in $60$ hours, all $3$ together would eat $15 + 12 + 10 = 37$ sheep.
So if takes them $60$ hours to eat $37$ sheep, it takes then $\dfrac {37} {60}$ hours to consume $1$ sheep.
Hence the result.
$\blacksquare$
Historical Note
As pointed out by John Fauvel and Jeremy Gray in their The History of Mathematics: A Reader of $1987$, this is a "more gory version" of the cistern problem, as seen for example in the Greek Anthology Book $\text {XIV}$, no. $132$.
Instead of $3$ pipes pouring water into a pool at given rates, $3$ animals remove flesh from a sheep at different rates.
Sources
- 1202: Leonardo Fibonacci: Liber Abaci
- 1987: John Fauvel and Jeremy Gray: The History of Mathematics: A Reader: $\text {7.A}$: The Thirteenth and Fourteenth Centuries: $\text {7.A1}$: Leonardo Fibonacci: $\text {(d)}$: The Lion, the leopard and the bear
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Liber Abaci: $89$