Two trains started at the same time , one from A to B and the other from B to A . If they arrived at B and A respectively 4 hours and 9 hours after they passed each other , the ratio of the speeds of the two trains was

Options

A. `2 : 1`
B. ` 3 : 2`
C. `4 : 3`
D. `5 : 4`

Answer

To solve the problem of finding the ratio of the speeds of two trains that started simultaneously from points A and B, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let Train A travel from A to B.
- Let Train B travel from B to A.
- After they pass each other, Train A takes 4 hours to reach B, and Train B takes 9 hours to reach A.

2. **Setting Up the Relationship**:
- The time taken by each train after they pass each other is inversely proportional to their speeds. This means that if one train takes longer to reach its destination, it is moving slower compared to the other train.

3. **Using Inverse Proportionality**:
- Let the speed of Train A be \( S_A \) and the speed of Train B be \( S_B \).
- According to the problem, we have:
\[ \frac{S_A}{S_B} = \frac{T_B}{T_A} \]
- Here, \( T_A = 4 \) hours (time taken by Train A after passing) and \( T_B = 9 \) hours (time taken by Train B after passing).

4. **Substituting the Values**:
- Substitute the times into the ratio:
\[ \frac{S_A}{S_B} = \frac{9}{4} \]

5. **Finding the Ratio of Speeds**:
- To express the ratio of speeds in a more conventional form, we can take the square root of the inverse of the time ratio:
\[ \frac{S_A}{S_B} = \sqrt{\frac{T_B}{T_A}} = \sqrt{\frac{9}{4}} = \frac{3}{2} \]

6. **Final Answer**:
- Therefore, the ratio of the speeds of the two trains is:
\[ \text{Ratio of speeds } S_A : S_B = 3 : 2 \]