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.