Answer:

Arthur could build the house in 60 days, and Peter could build it in 40 days.

Let x be the speed that Peter works, measured in houses per day.

Arthur works at a speed of (2/3) x.

So their total working speed is x + (2/3) x, which is (5/3) x.  Since they can build one house in 24 days, we know they can build 1/24 houses per day, working together.  In other words,

(5/3) x = 1/24

Equating the means and the extremes, we simplify this as follows:

(5/3) x = 1/24
5x / 3 = 1 / 24
3 = 120x
1/40 = x

So Peter can build 1/40 houses per day, or one house every 40 days.

Arthur works at 2/3 the speed of Peter, so he can build (2/3)(1/40) = 1/60 houses per day, or one house every 60 days.