A, B and C are three taps in a tank. Taps A and B can fill the tank independently in 20 min and 40 min, respectively. Tap C can empty the tank in 60 min. If all the three taps are opened together, how long will they take to fill the tank?

Options

A. 17 min
B. `17 1/2` min
C. `17 1/7` min
D. None of these

Answer

To solve the problem of how long it will take for taps A, B, and C to fill the tank when opened together, we will follow these steps:

### Step 1: Determine the filling rates of taps A and B, and the emptying rate of tap C.

- **Tap A** fills the tank in 20 minutes. Therefore, the rate of tap A is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{20 \text{ min}} = \frac{1}{20} \text{ tanks per minute} \]

- **Tap B** fills the tank in 40 minutes. Therefore, the rate of tap B is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{40 \text{ min}} = \frac{1}{40} \text{ tanks per minute} \]

- **Tap C** empties the tank in 60 minutes. Therefore, the rate of tap C (as it is emptying) is:
\[ \text{Rate of C} = -\frac{1 \text{ tank}}{60 \text{ min}} = -\frac{1}{60} \text{ tanks per minute} \]

### Step 2: Combine the rates of all three taps.

When all three taps are opened together, the net rate of filling the tank is the sum of the rates of taps A and B minus the rate of tap C:
\[ \text{Net Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
\[ \text{Net Rate} = \frac{1}{20} + \frac{1}{40} - \frac{1}{60} \]

### Step 3: Find a common denominator and calculate the net rate.

The least common multiple (LCM) of 20, 40, and 60 is 120. We will convert each rate to have a denominator of 120:
- For tap A:
\[ \frac{1}{20} = \frac{6}{120} \]
- For tap B:
\[ \frac{1}{40} = \frac{3}{120} \]
- For tap C:
\[ -\frac{1}{60} = -\frac{2}{120} \]

Now, substituting these values into the net rate equation:
\[ \text{Net Rate} = \frac{6}{120} + \frac{3}{120} - \frac{2}{120} = \frac{6 + 3 - 2}{120} = \frac{7}{120} \text{ tanks per minute} \]

### Step 4: Calculate the time taken to fill the tank.

To find the time taken to fill one tank, we take the reciprocal of the net rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\text{Net Rate}} = \frac{1}{\frac{7}{120}} = \frac{120}{7} \text{ minutes} \]

### Step 5: Convert the time into minutes and seconds.

Calculating \( \frac{120}{7} \):
\[ \frac{120}{7} \approx 17.14 \text{ minutes} \]
This can be expressed as:
- 17 minutes and \( 0.14 \times 60 \approx 8.57 \) seconds.

### Final Answer:
It will take approximately **17 minutes and 9 seconds** to fill the tank when all three taps are opened together.

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