Solution to the Problem:

The answers are (a) 66 2/3 eggs, (b) 3/2 days, (c) 3/2 chickens.

If a chicken and a half can lay an egg and a half in a day and a half, then one chicken can lay one egg in a day and a half. So one chicken can lay 2/3 of an egg in a day.
The problem that most people had with this was the time. They just divided to get one chicken laying one egg in one day. To see why this is not true, let's look at an easier example. Let's say that "two chickens can lay two eggs in two days." Then it takes one chicken two days to lay one egg -- not one day! If one chicken could lay one egg in one day, then ONE chicken could lay two eggs in two days.

(a): 1 chicken can lay 20/3 eggs in 10 days so
10 chickens can lay 10 times that amount in the same number of days, so
10 chickens can lay 200/3 or 66 2/3 eggs in 10 days.
10 chickens x 10 days x 2/3 egg/chicken/day = 66 2/3 eggs,

(b): 1 chicken can lay 2/3 egg in 1 day, so
10 chickens can lay 20/3 eggs in 1 day, so
10 chickens can lay 20/3x3/2 = 10 eggs in 3/2 days.
Twice as many chickens would lay twice as many eggs in the same amount of time,
so 10 chickens lay 10 eggs in 1 1/2 days,

(c): 1 chicken can lay 1 egg in 3/2 day, so
1 chicken can lay 20/3 eggs in 10 days, so
3/2 chickens can lay 10 eggs in 10 days.
1 1/2 chickens can lay an egg per day so they can lay 10 eggs in 10 days.