A tyre has 2 punctures. The first puncture alone would have made the tyre flat in 9 minutes and the second alone would have done it in 6 minutes. If air leaks out at a constant rate, how long does it take both the punctures together to make it flat?
एक टायर में दो पंक्चर हैं। पहला पंक्चर अकेला टायर को 9 मिनट में फ्लैट कर देता है और दूसरा पंक्चर अकेला वह 6 मिनट में कर देता है। यदि हवा स्थिर गति से निकले, तो दोनों पंक्चर मिल कर उसे कितनी देर में फ्लैट कर देंगे?

Options

A. `(1)/(2)` मिनट
B. `3(1)/(2)` मिनट
C. `3(3)/(5)` मिनट
D. `4(1)/(4)` मिनट

Explanation

To solve the problem of how long it takes for both punctures together to make the tyre flat, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Rate of Each Puncture:**
- The first puncture makes the tyre flat in 9 minutes. Therefore, its rate of leaking air is:
\[ \text{Rate of Puncture 1} = \frac{1}{9} \text{ tyre per minute} \]
- The second puncture makes the tyre flat in 6 minutes. Therefore, its rate of leaking air is:
\[ \text{Rate of Puncture 2} = \frac{1}{6} \text{ tyre per minute} \]

2. **Combine the Rates of Both Punctures:**
- To find the combined rate of both punctures working together, we add their rates:
\[ \text{Combined Rate} = \frac{1}{9} + \frac{1}{6} \]

3. **Find a Common Denominator:**
- The least common multiple (LCM) of 9 and 6 is 18. We can rewrite the fractions with a common denominator:
\[ \frac{1}{9} = \frac{2}{18} \quad \text{and} \quad \frac{1}{6} = \frac{3}{18} \]
- Now we can add them:
\[ \text{Combined Rate} = \frac{2}{18} + \frac{3}{18} = \frac{5}{18} \text{ tyre per minute} \]

4. **Calculate the Time Taken to Make the Tyre Flat:**
- To find the time taken for both punctures to make the tyre flat, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tyre}}{\frac{5}{18} \text{ tyre per minute}} = \frac{18}{5} \text{ minutes} \]
- Converting this into a mixed number:
\[ \frac{18}{5} = 3 \frac{3}{5} \text{ minutes} \]

### Final Answer:
The time taken for both punctures together to make the tyre flat is \( 3 \frac{3}{5} \) minutes or 3.6 minutes.

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