A and B are at a distance of $800\text{m}$. They start moving towards each other at $20\text{kmph}$ and $24\text{kmph}$ respectively. As they start, a bird sitting on the cap of A starts flying towards B, touches B and then returns towards A and so on, till they meet. What is the distance traveled by the bird, if its speed is $176\text{kmph}$?

Instead of tracking every single back-and-forth trip of the bird, we can solve this problem elegantly by finding out exactly how long the bird was flying (which is equal to the time taken by A and B to meet).
1. Convert initial separation distance to kilometers:

$$\text{Distance}=800\text{m}=\frac{800}{1000}=0.8\text{km}$$

2. Calculate Relative Speed of A and B:
Since they are walking towards each other, their relative speed is the sum of their speeds:

$$\text{Relative Speed}=20\text{kmph}+24\text{kmph}=44\text{kmph}$$

3. **Calculate Time taken by A and B to meet ($t$):**

$$t=\frac{\text{Distance}}{\text{Relative Speed}}=\frac{0.8\text{km}}{44\text{kmph}}=\frac{8}{440}=\frac{1}{55}\text{hours}$$

4. Calculate total distance covered by the bird:
The bird flies continuously at a constant speed of $176\text{kmph}$ during this entire duration $t$:

$$\text{Distance covered by bird}=\text{Speed of bird}\times t$$

$$\text{Distance}=176\times \frac{1}{55}=\frac{176}{55}\text{km}$$

Dividing numerator and denominator by $11$:

$$\text{Distance}=\frac{16}{5}=3.2\text{km}$$

5. Convert back to meters:

$$\text{Distance}=3.2\times 1000=3200\text{m}$$