A starts from a place P to go to a place Q. At the same time B starts from Q to P. If after meeting each other A and B took 16 and 25 hours more respectively to reach their destinations, the ratio of their speeds is:

Options

A. `3:2`
B. `5:4`
C. `9:4`
D. `9:13`

Answer

To solve the problem, we will use the relationship between speed, time, and distance. Let's denote the speeds of A and B as \( S_A \) and \( S_B \) respectively.

### Step-by-step Solution:

1. **Understanding the Problem**:
- A starts from point P to point Q.
- B starts from point Q to point P.
- After meeting, A takes 16 hours to reach Q, and B takes 25 hours to reach P.

2. **Using the Concept of Relative Speed**:
- When A and B meet, they have covered some distances. Let’s denote the distance A has left to reach Q after meeting as \( D_A \) and the distance B has left to reach P as \( D_B \).
- The time taken by A to cover \( D_A \) after meeting is 16 hours, and the time taken by B to cover \( D_B \) after meeting is 25 hours.

3. **Setting Up the Equations**:
- From the definition of speed, we know that:
\[ D_A = S_A \times 16 \]
\[ D_B = S_B \times 25 \]

4. **Using the Distance Relationship**:
- Since A and B meet at the same point, the distances they have left to travel are proportional to their speeds:
\[ \frac{D_A}{D_B} = \frac{S_A}{S_B} \]
- Substituting the expressions for \( D_A \) and \( D_B \):
\[ \frac{S_A \times 16}{S_B \times 25} = \frac{S_A}{S_B} \]

5. **Cross Multiplying**:
- Cross multiplying gives us:
\[ S_A \times 25 = S_B \times 16 \]

6. **Finding the Ratio of Speeds**:
- Rearranging the equation gives:
\[ \frac{S_A}{S_B} = \frac{16}{25} \]
- To express this in a more standard form, we can write:
\[ \frac{S_A}{S_B} = \frac{16}{25} = \frac{16 \div 4}{25 \div 4} = \frac{4}{6.25} \]
- This is equivalent to:
\[ \frac{S_A}{S_B} = \frac{4}{5} \]

7. **Final Ratio**:
- Thus, the ratio of their speeds \( S_A : S_B \) is \( 4 : 5 \).

### Conclusion:
The ratio of the speeds of A and B is \( 4 : 5 \).