Problem Solving

Hose A takes 2 days to fill a pool, hose B takes 3 days to fill a pool, and hose C takes 6 days to fill a pool. All three hoses are used to fill the first one-third of the pool, at which time hose B stops working and hose A and hose C continue until it is two-thirds full. At that point, hose C stops working and hose A continues until the pool is full. How long did it take to fill the entire pool?

a) 1.5
b) 2.5
c) 3.5
d) 4.5
e) 1.2

Anaira Mitch wrote:Hose A takes 2 days to fill a pool, hose B takes 3 days to fill a pool, and hose C takes 6 days to fill a pool. All three hoses are used to fill the first one-third of the pool, at which time hose B stops working and hose A and hose C continue until it is two-thirds full. At that point, hose C stops working and hose A continues until the pool is full. How long did it take to fill the entire pool?

a) 1.5
b) 2.5
c) 3.5
d) 4.5
e) 1.2

Let the pool = 6 liters.
Since A takes 2 days to fill the 6-liter pool, A's rate = w/t = 6/2 = 3 liters per day.
Since B takes 3 days to fill the 6-liter pool, B's rate = w/t = 6/3 = 2 liters per day.
Since C takes 6 days to fill the 6-liter pool, C's rate = w/t = 6/6 = 1 liter per day.

All three hoses are used to fill the first one-third of the pool.
Combined rate for A+B+C = 3+2+1 = 6 liters per day.
Time for A+B+C to fill 1/3 of the 6-liter pool = (1/3 of 6)/(combined rate for A+B+C) = 2/6 = 1/3 day.

Hose A and hose C continue until it is two-thirds full.
Combined rate for A+C = 3+1 = 4 liters per day.
Time for A+C to fill another 1/3 of the 6-liter pool = (1/3 of 6)/(combined rate for A+C) = 2/4 = 1/2 day.

Hose A continues until the pool is full.
Time for A to fill the remaining 1/3 of the 6-liter pool = (1/3 of 6)/(A's rate) = 2/3 day.

Total time = (1/3 day) + (1/2 day) + (2/3 day) = 1.5 days.

The correct answer is A.