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 to reach at M?

Options

A. `1:4`
B. `1:1`
C. `4:5`
D. `16:25`

Answer

To find the ratio of the time taken by person P and person Q to reach point M, we can use the relationship between speed, distance, and time.

### Step-by-Step Solution:

1. **Identify the Speeds**:
- Speed of P (SP) = 16 km/h
- Speed of Q (SQ) = 25 km/h

2. **Understanding the Relationship**:
- The time taken by each person to reach point M can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- Since they start at the same time and meet at point M, we can denote:
- Time taken by P to reach M = T1
- Time taken by Q to reach M = T2

3. **Using the Formula**:
- Let the distance traveled by P to reach M be D1 and the distance traveled by Q to reach M be D2.
- Therefore, we can write:
\[ T1 = \frac{D1}{SP} \quad \text{and} \quad T2 = \frac{D2}{SQ} \]

4. **Finding the Ratio of Times**:
- We want to find the ratio \( \frac{T1}{T2} \):
\[ \frac{T1}{T2} = \frac{\frac{D1}{SP}}{\frac{D2}{SQ}} = \frac{D1 \cdot SQ}{D2 \cdot SP} \]

5. **Using the Concept of Relative Distances**:
- Since both P and Q are moving towards each other, the distances D1 and D2 are proportional to their speeds. Thus:
\[ \frac{D1}{D2} = \frac{SQ}{SP} \]
- Therefore, we can substitute this into our ratio:
\[ \frac{T1}{T2} = \frac{SQ}{SP} = \frac{25}{16} \]

6. **Final Ratio**:
- Thus, the ratio of the time taken by P to the time taken by Q to reach point M is:
\[ \frac{T1}{T2} = \frac{25}{16} \]

### Conclusion:
The ratio of the time taken by P and Q to reach point M is \( 25:16 \).