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Inlet Pipe A can fill a cistem in 35 hou...

Inlet Pipe A can fill a cistem in 35 hours while outlet Pipe B can drain the filled cistern in 40 hours. The two pipes are opened together when the cistem is empty, but the outlet pipe is closed when the cistern is three-fifths full. How many hours did it take in all to fill the cistern?
A)185
B)182
C)184
D)180

A

185

B

182

C

184

D

180

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine how long it takes for the inlet pipe A and outlet pipe B to fill the cistern together until it reaches three-fifths full, and then calculate the remaining time to fill the cistern completely after closing the outlet pipe. ### Step-by-Step Solution: 1. **Determine the rates of the pipes:** - Inlet Pipe A can fill the cistern in 35 hours. Therefore, its rate is: \[ \text{Rate of A} = \frac{1}{35} \text{ cisterns per hour} \] - Outlet Pipe B can drain the cistern in 40 hours. Therefore, its rate is: \[ \text{Rate of B} = \frac{1}{40} \text{ cisterns per hour} \] 2. **Calculate the combined rate when both pipes are open:** - When both pipes are open, the effective rate is: \[ \text{Combined Rate} = \text{Rate of A} - \text{Rate of B} = \frac{1}{35} - \frac{1}{40} \] - To combine these fractions, find a common denominator (which is 280): \[ \text{Combined Rate} = \frac{8}{280} - \frac{7}{280} = \frac{1}{280} \text{ cisterns per hour} \] 3. **Determine how long it takes to fill three-fifths of the cistern:** - To fill three-fifths of the cistern, we can use the formula: \[ \text{Time} = \frac{\text{Volume}}{\text{Rate}} = \frac{3/5}{1/280} = \frac{3 \times 280}{5} = 168 \text{ hours} \] 4. **Calculate the remaining volume to fill the cistern:** - After three-fifths of the cistern is filled, two-fifths remain: \[ \text{Remaining Volume} = 1 - \frac{3}{5} = \frac{2}{5} \] 5. **Determine how long it takes to fill the remaining volume with only the inlet pipe A:** - Since the outlet pipe B is closed, only the inlet pipe A is filling the cistern: \[ \text{Time to fill remaining} = \frac{2/5}{1/35} = \frac{2 \times 35}{5} = 14 \text{ hours} \] 6. **Calculate the total time taken to fill the cistern:** - The total time taken is the sum of the time to fill three-fifths and the time to fill the remaining two-fifths: \[ \text{Total Time} = 168 + 14 = 182 \text{ hours} \] ### Final Answer: The total time taken to fill the cistern is **182 hours**.
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