Question

Two pipes A and B would fill a cistern in 30 and 40 minutes respectively. Both pipes are opened. Determine the time the first pipe can be turned off so that the cistern may be just filled in 20 minutes?

• A. After 10 minutes
• B. After 15 minutes
• C. After 20 minutes
• D. After 22 minutes
• E. After 25 minutes

Answer 1

The Correct Option is B

To determine the time after which pipe A can be turned off such that the cistern is filled in exactly 20 minutes, we first need to calculate the work done by each pipe.

1. Pipe A can fill the cistern in 30 minutes, hence its rate of work is \(\frac{1}{30}\) per minute.
2. Pipe B can fill the cistern in 40 minutes, so its rate of work is \(\frac{1}{40}\) per minute.
3. Both pipes together fill the cistern at a rate of: \(\frac{1}{30} + \frac{1}{40} = \frac{4 + 3}{120} = \frac{7}{120}\) per minute.
4. If both pipes are open for 20 minutes, the total combined work done would be: \(20 \times \frac{7}{120} = \frac{7}{6}\), which exceeds the full cistern (1). Therefore, pipe A must be closed earlier.
5. Let pipe A work for x minutes. Then, the work done by pipe A is \(\frac{x}{30}\), and by pipe B for 20 minutes it is \(\frac{20}{40} = \frac{1}{2}\).
6. For the cistern to be just full, the equation would be: \(\frac{x}{30} + \frac{1}{2} = 1\).
7. Solving this equation:
   • \(\frac{x}{30} = 1 - \frac{1}{2} = \frac{1}{2}\)
   • \(x = \frac{30}{2} = 15\) minutes

Therefore, pipe A should be turned off after 15 minutes for the cistern to be completely filled in 20 minutes when both pipes are initially turned on. Therefore, the correct answer is "After 15 minutes".

• Elimination of other options:
  • After 10 minutes: After 10 minutes with both pipes, the cistern will be \(10 \times \frac{7}{120} = \frac{7}{12}\) full. With only pipe B running afterwards, it would not fill the cistern in the remaining 10 minutes.
  • After 20 minutes: This cannot be the correct choice because it implies pipe A works for the entire duration.
  • After 22 minutes and After 25 minutes: These times would overfill the cistern when combined with the performance of pipe B.