Tap A can fill a tank in 6 h, whereas Tap B can fill it in 8 h. Tap C can empty the same tank in 4 h. If all the taps are opened together, how long will it take to fill the tank?

Correct option is D

Given:

Tap A fills the tank in 6 hours

Tap B fills the tank in 8 hours

Tap C empties the tank in 4 hours

All taps are opened together

Solution:

Tap A's filling rate = $$\frac{1}{6} \text{ tank/hour}$$​​

Tap B's filling rate = $$\frac{1}{8} \text{ tank/hour}$$​​

Tap C's emptying rate = $$\frac{1}{4} \text{ tank/hour}$$​​

Net Rate = $$\frac{1}{6} + \frac{1}{8} - \frac{1}{4}.$$​​

Net Rate = $$\frac{4}{24} + \frac{3}{24} - \frac{6}{24} = \frac{1}{24} \text{ tank/hour}.$$​​

Time = $$\frac{1}{\frac{1}{24}} = 24 \text{ hours}.$$​​

Alternate Solution:

LCM of 6, 8, and 4 = 24 units (total capacity of the tank).

Tap A's efficiency = $$\frac{24 }{6 } = 4 \text{ units/hour (fills)}$$​​

Tap B's efficiency = $$\frac{24 }{8 } = 3 \text{ units/hour (fills)}$$​​

Tap C's efficiency = $$\frac{24 }{4 } = 6 \text{ units/hour (empties)}$$​​

Net Efficiency = $$4 \text{ (A)} + 3 \text{ (B)} - 6 \text{ (C)} = 1 \text{ unit/hour (fills)}.$$​​

Time = $$\frac{\text{Total Work}}{\text{Net Efficiency}} = \frac{24 \text{ units}}{1 \text{ unit/hour}} = 24 \text{ hours}.$$​​