Solving a work problem using a rational equation

Question

For a certain bathtub, it takes the hot water faucet 3 times as long to fill the tub as it does the cold water faucet. Left on together, the two faucets take 6 minutes to fill the tub. How long will it take the hot water faucet to fill the tub by itself?

Answer 1

Since the hot water faucet takes three times as long to fill the tub, let t represent the time for the cold water to fill the tub alone and 3t reresent the time for the hot water faucet to fill the tub alone. Therefore the rate of the hot water faucet is 1/(3t) and the rate of the cold water faucet is 1/t. Since the rate times the time equals one job done, rt=1, then 1/(3t)x6 + (1/t)x6 = 1 and solve for t. Multiply both sides of the equation by the common denominator of 3t which produces t=8 which means the cold water faucet will fill the in 8 minutes and the hot water faucet will fill the tub three times as long or 24 minutes.

Answer 2

x is the work rate of the cold water faucet.

3x is the work rate of the hot water faucet.

6 is hte combined workrate.

the cold water faucet fills 1/x of the tub per minute.

the hot water faucet fills 1/(3x) of the tub per minute.

working together they fill 1/6 of the tub per minute.

combined speed is

1/x + 1/(3x) = 1/6

6 + 2 = x <--- multiplies everything by 6x

x = 8

The cold water faucet takes 8 minutes to fill the tub.

The hot water faucet takes 24 minutes to fill the tub.

The cold water faucet fills 1/8 of the tub per minute.

The hot water faucet fills 1/24 of the tub per minute.

Working together they fill 1/8 + 1/24 = 3/24 + 1/24 = 4/24 = 1/6 of the tub per minute.

Working together they fill the tub in 6 minutes, as given.