Pipes X and Y can fill a tank in 30 minutes and 45 minutes, respectively. whereas pipe Z can empty the full tank in 1 hour. Pipes X and Y are opened together for 10 minutes. Then, pipe X is closed, Z is opened instantly and Y continued to fill. The total time (from the beginning) taken to fill the tank is:

Options

A. 80 minutes
B. 100 minutes
C. 75 minutes
D. 90 minutes

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the rates of filling and emptying for each pipe.
- Pipe X can fill the tank in 30 minutes, so its rate is \( \frac{1}{30} \) of the tank per minute.
- Pipe Y can fill the tank in 45 minutes, so its rate is \( \frac{1}{45} \) of the tank per minute.
- Pipe Z can empty the tank in 60 minutes, so its rate is \( -\frac{1}{60} \) of the tank per minute (negative because it is emptying).

### Step 2: Calculate the combined rate of pipes X and Y.
The combined rate of pipes X and Y when both are open is:
\[ \text{Rate of X} + \text{Rate of Y} = \frac{1}{30} + \frac{1}{45} \]
To add these fractions, we need a common denominator. The least common multiple of 30 and 45 is 90.
\[ \frac{1}{30} = \frac{3}{90}, \quad \frac{1}{45} = \frac{2}{90} \]
Thus,
\[ \text{Combined rate} = \frac{3}{90} + \frac{2}{90} = \frac{5}{90} = \frac{1}{18} \text{ of the tank per minute.} \]

### Step 3: Calculate the amount of tank filled in the first 10 minutes.
In 10 minutes, the amount of tank filled by pipes X and Y is:
\[ \text{Amount filled} = \text{Rate} \times \text{Time} = \frac{1}{18} \times 10 = \frac{10}{18} = \frac{5}{9} \text{ of the tank.} \]

### Step 4: Determine the remaining part of the tank to be filled.
The remaining part of the tank after 10 minutes is:
\[ 1 - \frac{5}{9} = \frac{4}{9} \text{ of the tank.} \]

### Step 5: Calculate the combined rate of pipes Y and Z.
When pipe X is closed and pipe Z is opened, the combined rate of pipes Y and Z is:
\[ \text{Rate of Y} + \text{Rate of Z} = \frac{1}{45} - \frac{1}{60} \]
Again, we find a common denominator, which is 180.
\[ \frac{1}{45} = \frac{4}{180}, \quad \frac{1}{60} = \frac{3}{180} \]
Thus,
\[ \text{Combined rate} = \frac{4}{180} - \frac{3}{180} = \frac{1}{180} \text{ of the tank per minute.} \]

### Step 6: Calculate the time taken to fill the remaining part of the tank.
To fill the remaining \( \frac{4}{9} \) of the tank at a rate of \( \frac{1}{180} \):
\[ \text{Time} = \frac{\text{Remaining part}}{\text{Rate}} = \frac{\frac{4}{9}}{\frac{1}{180}} = \frac{4}{9} \times 180 = 80 \text{ minutes.} \]

### Step 7: Calculate the total time taken to fill the tank.
The total time taken to fill the tank is:
\[ 10 \text{ minutes (initial)} + 80 \text{ minutes (remaining)} = 90 \text{ minutes.} \]

### Final Answer:
The total time taken to fill the tank is **90 minutes**.

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