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.  


To solve this problem, we need to calculate how much total distance the fly covers until the two mathematicians, K and J, meet. 

Here's a step-by-step explanation:

### Step 1: Determine when K and J meet.
K walks at 5 km/h and J walks at 4 km/h. Since they are walking towards each other, their combined speed is:
\[
5 + 4 = 9 \text{ km/h}
\]

The initial distance between them is 1 km. So, the time taken for them to meet is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1}{9} \text{ hours}
\]

### Step 2: Calculate the distance the fly covers.
The fly is flying back and forth between K and J at a speed of 20 km/h. The fly flies continuously for the same amount of time it takes for K and J to meet, which is \(\frac{1}{9}\) hours. The total distance the fly covers is:
\[
\text{Distance covered by fly} = \text{Fly's speed} \times \text{Time} = 20 \times \frac{1}{9} = \frac{20}{9} \text{ km}
\]

Thus, the total distance covered by the fly is approximately:
\[
\boxed{2.22 \text{ km}}
\]

### Summary:
The fly covers a total distance of \(\frac{20}{9} \approx 2.22\) km before K and J meet.

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Do you want any more details or have any questions?  
Here are 5 related questions:
1. What if the fly's speed was 30 km/h instead of 20 km/h?
2. How would the problem change if K and J were 2 km apart?
3. What if K walked at 6 km/h and J at 3 km/h?
4. How much distance would the fly cover if it started from J’s head instead?
5. What if the road was not straight but had a curve?

**Tip:** When objects move towards each other, their relative speed is the sum of their individual speeds.

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.

In this problem, two people start walking towards each other, and a fly moves back and forth between them. Learn how to calculate the total distance covered by the fly using relative speed and uniform motion concepts. Suitable for students in Grades 9-10.

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