Two taps can fill a tank in 15 and 12 minutes respectively. A third tap can empty it in 20 minutes. If all the taps are opened at the same time, then in how much time will the tank be filled ?

Answer

To solve the problem of how long it will take to fill the tank when all three taps are opened simultaneously, we can follow these steps:

### Step 1: Determine the rate of each tap
- **First tap** fills the tank in 15 minutes. Therefore, the rate of the first tap is:
\[ \text{Rate of first tap} = \frac{1}{15} \text{ tank per minute} \]

- **Second tap** fills the tank in 12 minutes. Therefore, the rate of the second tap is:
\[ \text{Rate of second tap} = \frac{1}{12} \text{ tank per minute} \]

- **Third tap** empties the tank in 20 minutes. Therefore, the rate of the third tap (which is negative since it empties) is:
\[ \text{Rate of third tap} = -\frac{1}{20} \text{ tank per minute} \]

### Step 2: Combine the rates of all taps
Now, we can add the rates of the first two taps and subtract the rate of the third tap:
\[ \text{Combined rate} = \frac{1}{15} + \frac{1}{12} - \frac{1}{20} \]

### Step 3: Find a common denominator
To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 15, 12, and 20 is 60.

- Convert each rate:
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{12} = \frac{5}{60}, \quad -\frac{1}{20} = -\frac{3}{60} \]

### Step 4: Add the rates
Now we can combine the rates:
\[ \text{Combined rate} = \frac{4}{60} + \frac{5}{60} - \frac{3}{60} = \frac{4 + 5 - 3}{60} = \frac{6}{60} = \frac{1}{10} \]

### Step 5: Calculate the time to fill the tank
The combined rate of \(\frac{1}{10}\) tank per minute means that the tank will be filled in:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{10} \text{ tank per minute}} = 10 \text{ minutes} \]

### Final Answer
Thus, the tank will be filled in **10 minutes**.

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