Q1. Two Mathematicians K and J, 1 km apart on a straight road start walking towards each other with constant speeds of 5 km/hour and 4 km/hour respectively, A fly, which flies at a constant speed of 20 km/hour is originally on K’s head. It flies towards J and after touching J’s Head immediately reverses its direction and flies towards K, then again towards J and so on. Find how much total distance the fly covers (in km) till K and J meet.  
*
20/11
2.11
2.22
10/3
None of the above

To solve this problem, we need to determine the total distance that the fly covers as K and J walk towards each other until they meet.

### Step-by-Step Solution:

1. **Relative Speed of K and J:**
   - Speed of K = 5 km/hour
   - Speed of J = 4 km/hour
   - Since they are moving towards each other, their relative speed is the sum of their speeds:
     \[
     \text{Relative speed} = 5 + 4 = 9 \text{ km/hour}
     \]

2. **Time Taken for K and J to Meet:**
   - They start 1 km apart.
   - Time taken to meet = Distance / Relative speed:
     \[
     \text{Time} = \frac{1 \text{ km}}{9 \text{ km/hour}} = \frac{1}{9} \text{ hours}
     \]

3. **Distance Covered by the Fly:**
   - Speed of the fly = 20 km/hour
   - The fly keeps flying back and forth between K and J until they meet.
   - The total time the fly spends flying is the same as the time K and J take to meet, which is \(\frac{1}{9}\) hours.
   - Therefore, the total distance the fly covers is:
     \[
     \text{Distance} = \text{Speed of the fly} \times \text{Time} = 20 \text{ km/hour} \times \frac{1}{9} \text{ hours} = \frac{20}{9} \text{ km} \approx 2.22 \text{ km}
     \]

So, the total distance covered by the fly is \(\frac{20}{9}\) km, which matches the answer choice:

**2.22 km**.

Thus, the correct answer is:

**Option 3: 2.22**

Would you like further details or explanations? Here are some related questions to explore:

1. What would the total distance be if the speeds of K and J were different?
2. How would the answer change if the fly's speed was 10 km/hour instead of 20 km/hour?
3. Can this problem be solved using an infinite series approach?
4. How does the fly's distance change if K and J start from different distances?
5. What happens if K and J start walking away from each other?

**Tip:** When dealing with problems involving objects moving towards each other, their relative speed is simply the sum of their individual speeds.

Discover how to calculate the total distance covered by a fly moving between two walkers on a straight road. This problem involves calculating relative speeds and time-distance relationships. Suitable for high school students learning about motion and speed in mathematics.

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