Three pipes A, B and C can fill a cistern in 15, 24 and 36 minutes respectively. If pipe D can drain a full tank in 1 hour, how long will it take for the tank to be filled if all the four pipes are kept open together?

Options

A. `9 1/8` hours
B. `5 (12)/(25)` hours
C. `8 (16)/(43)` hours
D. `7 2/3` hours

Answer

To solve the problem, we need to determine how long it will take to fill a tank when all four pipes A, B, C, and D are opened together. Here’s a step-by-step solution:

### Step 1: Determine the filling rates of pipes A, B, and C.
- Pipe A fills the tank in 15 minutes, so its rate is:
\[ \text{Rate of A} = \frac{1}{15} \text{ tanks per minute} \]

- Pipe B fills the tank in 24 minutes, so its rate is:
\[ \text{Rate of B} = \frac{1}{24} \text{ tanks per minute} \]

- Pipe C fills the tank in 36 minutes, so its rate is:
\[ \text{Rate of C} = \frac{1}{36} \text{ tanks per minute} \]

### Step 2: Determine the draining rate of pipe D.
- Pipe D drains the tank in 60 minutes (1 hour), so its rate is:
\[ \text{Rate of D} = -\frac{1}{60} \text{ tanks per minute} \]
(Note: The negative sign indicates that it is draining the tank.)

### Step 3: Calculate the combined rate of all pipes.
Now, we will add the rates of pipes A, B, C, and D together:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} + \text{Rate of D} \]
Substituting the rates we calculated:
\[ \text{Combined Rate} = \frac{1}{15} + \frac{1}{24} + \frac{1}{36} - \frac{1}{60} \]

### Step 4: Find a common denominator.
The least common multiple (LCM) of 15, 24, 36, and 60 is 360. We will convert each rate to have this common denominator:
- For A:
\[ \frac{1}{15} = \frac{24}{360} \]
- For B:
\[ \frac{1}{24} = \frac{15}{360} \]
- For C:
\[ \frac{1}{36} = \frac{10}{360} \]
- For D:
\[ -\frac{1}{60} = -\frac{6}{360} \]

### Step 5: Combine the rates.
Now we can combine these fractions:
\[ \text{Combined Rate} = \frac{24}{360} + \frac{15}{360} + \frac{10}{360} - \frac{6}{360} = \frac{43}{360} \text{ tanks per minute} \]

### Step 6: Calculate the time to fill the tank.
To find the time taken to fill the tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{43}{360} \text{ tanks per minute}} = \frac{360}{43} \text{ minutes} \]

### Step 7: Convert the time into hours and minutes.
Calculating \( \frac{360}{43} \):
- Dividing gives approximately \( 8.37 \) minutes.
- This can be converted to minutes and seconds:
- \( 8 \) minutes and \( 0.37 \times 60 \approx 22.2 \) seconds.

Thus, the time taken to fill the tank is approximately:
\[ 8 \text{ minutes and } 22 \text{ seconds} \]

### Final Answer:
The tank will be filled in approximately 8 minutes and 22 seconds when all four pipes are opened together.

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