A persons P starts his journey from A and another person Q starts his journey from B, towards each other. The speeds of P and Q are 16 km/h and 25 km/h respectivley and they meet at point M somewhere between A and B when they start their journey simultaneously.
What is the ratio of time taken by P and Q after meeting each other at M to reach B and A respectively?

Options

A. `25:16`
B. `625:256`
C. `16:25`
D. `4:5`

Answer

To solve the problem, we need to find the ratio of the time taken by person P to reach point B after meeting at point M and the time taken by person Q to reach point A after meeting at point M.

### Step-by-Step Solution:

1. **Identify the Speeds**:
- Speed of P (from A to B) = 16 km/h
- Speed of Q (from B to A) = 25 km/h

2. **Understanding the Meeting Point**:
- When P and Q meet at point M, they have been traveling for the same amount of time. Let’s denote the time taken to meet as \( t \).

3. **Distance Covered Until Meeting**:
- Distance covered by P until M = Speed of P × Time = \( 16t \)
- Distance covered by Q until M = Speed of Q × Time = \( 25t \)

4. **Total Distance Between A and B**:
- The total distance between A and B = Distance covered by P + Distance covered by Q = \( 16t + 25t = 41t \)

5. **Time Taken After Meeting**:
- After meeting, P will travel the remaining distance to B, and Q will travel the remaining distance to A.
- Remaining distance for P to reach B = Total distance - Distance covered by P = \( 41t - 16t = 25t \)
- Remaining distance for Q to reach A = Total distance - Distance covered by Q = \( 41t - 25t = 16t \)

6. **Calculating Time Taken After Meeting**:
- Time taken by P to reach B after meeting = Remaining distance / Speed of P = \( \frac{25t}{16} \)
- Time taken by Q to reach A after meeting = Remaining distance / Speed of Q = \( \frac{16t}{25} \)

7. **Finding the Ratio of Time Taken**:
- The ratio of time taken by P to reach B and Q to reach A after meeting is given by:
\[ \text{Ratio} = \frac{\text{Time taken by P}}{\text{Time taken by Q}} = \frac{\frac{25t}{16}}{\frac{16t}{25}} = \frac{25t}{16} \times \frac{25}{16t} = \frac{625}{256} \]

8. **Final Answer**:
- The ratio of time taken by P and Q after meeting at M to reach B and A respectively is \( 625 : 256 \).