A cistern can be filled by a tap in 6 hours and emptied by an outlet pipe in 7.5 hours.How long will it take to fill the cistern if both the tap and the pipe are opened together?

Correct option is D

Given:

Time taken by tap to fill the cistern = 6 hours

Time taken by outlet pipe to empty the cistern = 7.5 hours

Formula Used:

Net rate = (Filling rate) - (Emptying rate)

Time = $$\frac1 {\text{ Net rate}}$$​​

Solution:

Filling rate = $$\frac{1}{6}$$​​

Emptying rate = $$\frac{1}{7.5} = \frac{2}{15}$$​​

Net rate = $$\frac{1}{6} - \frac{2}{15}$$​​

​$$= \frac{5 - 4}{30} = \frac{1}{30}$$​​

Time to fill the cistern = $$\frac{1}{1/30}$$​​

= 30 hours

Alternate Solution:

Total Work = Time $$\times$$​ Net Work Rate

total capacity = LCM (6,7) = 30 units

Tap fills = $$\frac{30}{6}$$​ = 5 units/hour

Pipe empties = $$\frac{30}{7.5} = \frac{30 \times 2}{15}$$​ = 4 units/hour

Net work rate = 5 - 4 = 1 unit/hour

Time = $$\frac{30}{1}$$​ = 30 hours