The problem states:

A cistern can be filled up by a tap in 6 hours. It is emptied by an outlet tap in 8 hours. If both the taps are opened, the cistern will be filled up in:

Solution:

Step 1: Determine the rate of filling and emptying.

• Filling tap: The cistern is filled in 6 hours, so the filling rate = $\frac{1}{6}$ cistern per hour.
• Emptying tap: The cistern is emptied in 8 hours, so the emptying rate = $\frac{1}{8}$ cistern per hour.

Step 2: Combine the rates when both taps are open.

When both taps are open, the net rate of filling is: $\text{Net rate} = \text{Filling rate} - \text{Emptying rate}$ $\text{Net rate} = \frac{1}{6} - \frac{1}{8}$

To subtract the fractions, find the least common denominator (LCD) of 6 and 8, which is 24: $\frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24}$ $\text{Net rate} = \frac{4}{24} - \frac{3}{24} = \frac{1}{24}$

Thus, the net rate of filling is $\frac{1}{24}$ cistern per hour.

Step 3: Calculate the time to fill the cistern.

If the cistern is filled at a rate of $\frac{1}{24}$ cistern per hour, the time to fill it completely is: $\text{Time} = \frac{1}{\text{Net rate}} = \frac{1}{\frac{1}{24}} = 24 \, \text{hours.}$

Final Answer:

The cistern will be filled in 24 hours.

Would you like a breakdown of any particular step?

Related Questions:

1. What happens if the emptying tap is faster than the filling tap?
2. How can we solve this type of problem using ratios instead of fractions?
3. If the filling rate changes to 4 hours, how does the solution change?
4. What if the outlet tap is opened only after 3 hours of filling? How long will it take then?
5. How would you approach this if the cistern's total capacity was provided explicitly?

Tip:

Always convert word problems into equations step-by-step to ensure clarity and avoid errors.