There are taps to fill a tank and a third to empty it. When the third tap is closed, they can fill the tank in 10 minutes and 12 minutes respectively. If all the three taps are opened, the tank is filled in 15 minutes. If the first two taps are closed, in what time can the third tap empty the tank when it is full ?

Answer

To solve the problem step by step, we will first determine the rates at which each tap fills or empties the tank, and then use these rates to find the time taken by the third tap to empty the tank.

### Step 1: Determine the rates of the taps
- The first tap fills the tank in 10 minutes, so its rate is:
\[ \text{Rate of Tap 1} = \frac{1}{10} \text{ tank per minute} \]

- The second tap fills the tank in 12 minutes, so its rate is:
\[ \text{Rate of Tap 2} = \frac{1}{12} \text{ tank per minute} \]

- Let the time taken by the third tap to empty the tank be \( x \) minutes. Therefore, its rate is:
\[ \text{Rate of Tap 3} = -\frac{1}{x} \text{ tank per minute} \quad (\text{negative because it empties the tank}) \]

### Step 2: Combine the rates when all taps are open
When all three taps are opened, they can fill the tank in 15 minutes. Thus, the combined rate is:
\[ \text{Combined Rate} = \frac{1}{15} \text{ tank per minute} \]

### Step 3: Set up the equation
The combined rate of the three taps can be expressed as:
\[ \frac{1}{10} + \frac{1}{12} - \frac{1}{x} = \frac{1}{15} \]

### Step 4: Solve for \( \frac{1}{x} \)
To solve for \( \frac{1}{x} \), we first find a common denominator for the fractions on the left side. The least common multiple (LCM) of 10, 12, and 15 is 60.

- Convert each term:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]

- Substitute these into the equation:
\[ \frac{6}{60} + \frac{5}{60} - \frac{1}{x} = \frac{4}{60} \]

### Step 5: Combine the fractions
Combine the fractions on the left:
\[ \frac{11}{60} - \frac{1}{x} = \frac{4}{60} \]

### Step 6: Isolate \( \frac{1}{x} \)
Rearranging gives:
\[ \frac{1}{x} = \frac{11}{60} - \frac{4}{60} = \frac{7}{60} \]

### Step 7: Solve for \( x \)
Taking the reciprocal gives:
\[ x = \frac{60}{7} \text{ minutes} \]

### Step 8: Convert to minutes and seconds
To convert \( \frac{60}{7} \) into minutes and seconds:
- \( 60 \div 7 = 8.57 \) minutes
- The integer part is 8 minutes.
- To find the seconds, calculate \( 0.57 \times 60 \approx 34.29 \) seconds.

Thus, the third tap takes approximately **8 minutes and 34 seconds** to empty the tank when it is full.

### Final Answer
The third tap can empty the tank in **8 minutes and 34 seconds**.

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