A chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days?

the farmer needs 3 hens to produce 12 eggs in 6 days

This is a classic problem that many people get wrong because they reason that half of a hen cannot lay an egg, and a hen cannot lay half an egg. However, we can get a satisfactory solution by treating this as a purely mathematical problem where the numbers represent averages.

To solve the problem, we first need to find the rate at which the hens lay eggs. The problem can be represented by the following equation, where RATE is the number of eggs produced per hen·day:

1½ hens × 1½ days × RATE = 1½ eggs

We convert this to fractions thus:

3/2 hens × 3/2 days × RATE = 3/2 eggs

Multiplying both sides of the equation by 2/3, we get:

1 hen × 3/2 days × RATE = 1 egg

Multiplying both sides of the equation again by 2/3 and solving for RATE, we get:

RATE = 2/3 eggs per hen·day

Now that we know the rate at which hens lay eggs, we can calculate how many hens (H) can produce 12 eggs in six days using the following equation:

H hens × 6 days × 2/3 eggs per hen·day = 12 eggs

Solving for H, we get:

H = 12 eggs /(6 days × 2/3 eggs per hen·day) = 12/4 = 3 hens

Therefore, the farmer needs 3 hens to produce 12 eggs in 6 days.