The following questions are accompanied by three statements A, B and C. You have to determine which statement(s) is/are necessary/sufficient to answer the question.
Find out the length of train A.
A. Train A crosses another train B moving in the opposite direction in 10 sec.
B. Ratio of the speeds of trains A and B is 1:2.
C. Length of train B is 25% more than that of train A.

Options

A. All the three together are not sufficient
B. Only A and C together
C. All the three together
D. Only A and B together

Answer

To determine the length of train A, we will analyze each statement provided (A, B, and C) and see how they contribute to finding the solution.

### Step-by-Step Solution:

1. **Understanding Statement A**:
- Statement A tells us that train A crosses train B, which is moving in the opposite direction, in 10 seconds.
- This implies that the combined length of both trains is crossed in this time.
- We can use the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
- However, we need to know the speed of the trains to find the distance.

**Hint**: Think about how crossing time relates to the lengths of the trains and their speeds.

2. **Understanding Statement B**:
- Statement B provides the ratio of the speeds of trains A and B, which is 1:2.
- Let’s denote the speed of train A as \( v \) and the speed of train B as \( 2v \).
- This information allows us to express the speeds in terms of a single variable.

**Hint**: Use the ratio of speeds to express one speed in terms of the other.

3. **Understanding Statement C**:
- Statement C states that the length of train B is 25% more than that of train A.
- If we let the length of train A be \( L_A \), then the length of train B can be expressed as:
\[ L_B = L_A + 0.25L_A = 1.25L_A \]
- This gives us a direct relationship between the lengths of the two trains.

**Hint**: Relate the lengths of the two trains using the percentage increase given.

4. **Combining the Information**:
- From Statement A, we have the time taken to cross each other (10 seconds).
- From Statement B, we have the speeds in terms of \( v \) and \( 2v \).
- From Statement C, we have the lengths of the trains in terms of \( L_A \).
- The total distance covered when the two trains cross each other can be expressed as:
\[ L_A + L_B = L_A + 1.25L_A = 2.25L_A \]
- The speed at which they cross each other is the sum of their speeds:
\[ v + 2v = 3v \]
- Therefore, we can set up the equation:
\[ 2.25L_A = 3v \times 10 \]
- Rearranging gives us:
\[ L_A = \frac{30v}{2.25} = \frac{30v \times 4}{9} = \frac{120v}{9} = \frac{40v}{3} \]
- This shows that we can find \( L_A \) if we know \( v \).

5. **Conclusion**:
- All three statements are necessary to find the length of train A.
- Statement A provides the crossing time, Statement B provides the speed ratio, and Statement C provides the relationship between the lengths of the trains.

### Final Answer:
All three statements (A, B, and C) together are necessary to determine the length of train A.