Filling a hot tub with a hose takes 4 hours. Draining the same tub takes 7 hours. If the drain is open, how long will it take to fill the tub

Question

 Filling a hot tub with a hose takes 4 hours. Draining the same tub takes 7 hours. If the drain is open, how long will it take to fill the tub? 

Answer 1

Hi Augstina,

 

takes 4 hours means 1/4 can be filled in 1 hour

takes 7 hours means 1/7 can be filled in 1 hour

1) by the time the tub is full both have been running for equal amount of time t.

2) the amount of water by the hose is equal to (1/4full)(1/hour)(t hour)

3) the amount of water by the drain is equal to (1/7full)(1/hour)(t hour) + 1full

 

1/4ft = 1/7ft + 1f    

1/4t = 1/7t + 1   //dividing both sides by f

 

t(1/4-1/7)=1

t=1/3/28

t=9.33hours

 

Answer 2

The rate at which the tub fills is 1 tub per 4 hours, or 1/4 tub per hour.

 

The rate at which it drains is 1 tub per 7 hours, or 1/7 tub per hour.

 

So if the it's being filled and drained at the same time, the rate at which is gains water is 1/4 - 1/7 tub per hour.

 

The least common denominator for this subtraction is 28, so the rate at which it gains water is 7/28 - 4/28 = 3/28.

 

So if the tub gains water at the rate of 3/28 tub per hour, then it would take 28/3 hours to fill one tub, which is 9 1/3 hours.