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# Eggs Per Hen-Day

There’s an old brain teaser that is posed as a word problem, something like this:

If a hen and a half lays an egg and a half in a day and a half, how many days will it take three hens to lay three eggs?

or

If a hen and a half lays an egg in a half in a day and a half, how many hens will it take to lay six eggs in six days?

These are essentially the same problem and there is no significant difference to “three” or “six”. I’ve seen several internet sites that go through the math to calculate the rate of egg laying to come up with the answer, but that is not necessary.

With a little knowledge of Algebra, we realize that this is a rate problem: how many eggs are produced per hen-day? Mathematically, we would write that this way:

 Eggs = Rate * Hens * Days

and then solving for “Rate” we get this:

 Rate = Eggs (Hens)(Days)

We then see immediately that we don’t even need to know Rate.

If we multiply Eggs and Hens by the same factor N:

 Rate = N * Eggs (N * Hens)(Days)

that factor, being in both the numerator and denominator, will cancel and Days must remain unchanged.

In the same way if we multiply Eggs and Days by the same factor N:

 Rate = N * Eggs (Hens)(N * Days)

again that factor, being in both the numerator and denominator, will cancel and Hens must remain unchanged.

So, we can change “one and a half” to X and three (or six) to Y:

If X hens lay X eggs in X days, how many days will it take for Y hens to lay Y eggs?

or

If X hens lay X eggs in X days, how many hens will it take to lay Y eggs in Y days?

For all values of X and Y, the answer will always be X.