A pipe can fill a tank in 10 h, while an another pipe can empty it in 6 h. Find the time taken to empty the tank, when both the pipes are opened simultaneously?

Options

A. 11h
B. 15h
C. 18h
D. 16h

Answer

To solve the problem, we need to determine the time taken to empty the tank when both the filling pipe and the emptying pipe are opened simultaneously. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the rates of the pipes
1. **Filling Pipe**: It can fill the tank in 10 hours.
- Rate of filling = \( \frac{1 \text{ tank}}{10 \text{ hours}} = 0.1 \text{ tanks/hour} \)

2. **Emptying Pipe**: It can empty the tank in 6 hours.
- Rate of emptying = \( \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tanks/hour} \)

### Step 2: Calculate the combined rate when both pipes are opened
- When both pipes are opened simultaneously, the filling pipe adds to the tank, while the emptying pipe takes away from it. Therefore, we can combine their rates:
- Combined rate = Rate of filling - Rate of emptying
- Combined rate = \( 0.1 - \frac{1}{6} \)

### Step 3: Convert the rates to a common denominator
- To perform the subtraction, we need a common denominator. The least common multiple (LCM) of 10 and 6 is 30.
- Convert \( 0.1 \) to a fraction: \( 0.1 = \frac{3}{30} \)
- Convert \( \frac{1}{6} \) to a fraction: \( \frac{1}{6} = \frac{5}{30} \)

### Step 4: Perform the subtraction
- Now we can subtract the rates:
- Combined rate = \( \frac{3}{30} - \frac{5}{30} = \frac{3 - 5}{30} = \frac{-2}{30} = -\frac{1}{15} \text{ tanks/hour} \)

### Step 5: Interpret the result
- The negative sign indicates that the tank is being emptied. Thus, the effective rate of emptying the tank is \( \frac{1}{15} \text{ tanks/hour} \).

### Step 6: Calculate the time to empty the tank
- If the rate of emptying is \( \frac{1}{15} \text{ tanks/hour} \), then the time taken to empty 1 tank is the reciprocal of the rate:
- Time = \( \frac{1 \text{ tank}}{\frac{1}{15} \text{ tanks/hour}} = 15 \text{ hours} \)

### Final Answer
- The time taken to empty the tank when both pipes are opened simultaneously is **15 hours**.

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