The following questions are accompanied by two statements (i) and (II). You have to determine which statements(s) is/are sufficient/ necessary to answer the questions.
Ratio of length of train - A and train - B is 4: 5. Find speed of train - A.
Statement I: Train - A can cross a 500m long platform in 28 seconds and train - A crosses train - B while running in same direction in 54 seconds.
Statement II: Train - B can cross a pole in 15 seconds.

Options

A. Statement (I) alone is sufficient to answer the question but statement (II) alone is not sufficient to answer the question.
B. Statement (II) alone is sufficient to answer the question but statement (I) alone is not sufficient to answer the question.
C. Both the statements taken together are necessary to answer the question, but neither of the statements alone is sufficient to answer the question.
D. Either statement (I) or statement (II) by itself is sufficient to answer the question.

Answer

To solve the problem of finding the speed of train A given the ratio of lengths of train A and train B and the two statements, we can follow these steps:

### Step 1: Understand the given information
We know that the ratio of the lengths of train A to train B is 4:5. We can represent the lengths of the trains as:
- Length of train A = 4x meters
- Length of train B = 5x meters

### Step 2: Analyze Statement I
Statement I provides two pieces of information:
1. Train A can cross a 500m long platform in 28 seconds.
2. Train A crosses train B while running in the same direction in 54 seconds.

From the first part, we can calculate the speed of train A:
- Speed of train A = (Length of platform + Length of train A) / Time taken
- Speed of train A = (500 + 4x) / 28

From the second part, we can derive the relationship between the speeds of train A and train B:
- Speed of train A - Speed of train B = (Length of train A + Length of train B) / Time taken
- Speed of train A - Speed of train B = (4x + 5x) / 54
- Speed of train A - Speed of train B = 9x / 54 = x / 6

### Step 3: Analyze Statement II
Statement II states that train B can cross a pole in 15 seconds. This gives us:
- Speed of train B = Length of train B / Time taken
- Speed of train B = 5x / 15 = x / 3

### Step 4: Combine the statements
Now we can use the information from both statements to find the speed of train A:
1. From Statement I, we have:
- Speed of train A = (500 + 4x) / 28
- Speed of train A - Speed of train B = x / 6

2. From Statement II, we have:
- Speed of train B = x / 3

Now we can substitute the speed of train B into the equation from Statement I:
- Speed of train A - (x / 3) = x / 6

Substituting the expression for Speed of train A:
- (500 + 4x) / 28 - x / 3 = x / 6

### Step 5: Solve for x
To solve for x, we can multiply through by the least common multiple of the denominators (which is 84) to eliminate the fractions:
- 84 * [(500 + 4x) / 28] - 84 * (x / 3) = 84 * (x / 6)

This simplifies to:
- 3 * (500 + 4x) - 28x = 14x
- 1500 + 12x - 28x = 14x
- 1500 = 14x + 16x
- 1500 = 30x
- x = 50

### Step 6: Calculate the speed of train A
Now that we have x, we can find the speed of train A:
- Speed of train A = x / 2 = 50 / 2 = 25 m/s

### Conclusion
Thus, the speed of train A is 25 m/s.