Question

There are two pipes leading to a tank. If pipe X alone is opened, the tank will be filled in 20 minutes. If pipe Y alone is opened, the tank will be filled in 25 minutes. Both pipes were opened together. After some time, pipe Y was closed. If it takes 15 minutes to fill the tank completely, then for how long is the pipe X left open

• A. 10 minutes 30 seconds
• B. 8 minute 45 second
• C. 7 minutes 15 seconds
• D. 9 minutes 40 seconds

Answer 1

Answer:

B. 8 minute 45 second

• The fundamental concept in time and work problems, including pipe and cisterns, is the 'rate of work'. This refers to the amount of work done per unit of time.

• If pipe X fills the tank in 20 minutes, its rate of filling is 1/20 of the tank per minute.

• Similarly, if pipe Y fills the tank in 25 minutes, its rate of filling is 1/25 of the tank per minute.

• When both pipes X and Y are opened together, their rates add up.

• Combined rate of X and Y = Rate of X + Rate of Y = (1/20 + 1/25) tank per minute.

• To add these fractions, find a common denominator, which is 100.

• (1/20 + 1/25) = (5/100 + 4/100) = 9/100 tank per minute.

• Let 't' be the time (in minutes) for which both pipes were opened together.

• In time 't', the work done by both pipes together is (9/100) * t.

• After time 't', pipe Y is closed, and pipe X continues to fill the remaining part of the tank.

• The total time taken to fill the tank is 15 minutes. This means pipe X was open for the entire 15 minutes.

• The portion of the tank filled by pipe X alone in the remaining time is (15 - t) minutes.

• The work done by pipe X alone in (15 - t) minutes is (1/20) * (15 - t).

• The sum of the work done by both pipes together and the work done by pipe X alone must equal the total work (filling 1 whole tank).

• Equation: (9/100) * t + (1/20) * (15 - t) = 1

1. Multiply the entire equation by 100 to eliminate denominators:

2. 9t + 5(15 - t) = 100

3. Expand the equation: 9t + 75 - 5t = 100

4. Simplify: 4t + 75 = 100

5. Isolate 't': 4t = 100 - 75

6. 4t = 25

7. Solve for 't': t = 25 / 4 minutes

• 't' represents the time both pipes were open together.

• t = 25/4 minutes = 6.25 minutes.

• The question asks for how long pipe X was left open after pipe Y was closed. This is the time when only pipe X was working.

• Time pipe X worked alone = Total time - Time both worked together

• Time pipe X worked alone = 15 minutes - t

• Time pipe X worked alone = 15 - 6.25 minutes = 8.75 minutes.

• Convert 0.75 minutes to seconds: 0.75 * 60 seconds = 45 seconds.

• Therefore, pipe X was left open for 8 minutes and 45 seconds after pipe Y was closed.