The sum of the length of two trains A and B is 660m. The ratio of the speed of A to that of B is 5 : 8. The ratio of time to cross an electric pole by train A to that by B is 4 :3. Find the difference between the length of the two trains.

Options

A. 50
B. 60
C. 80
D. 75

Answer

To solve the problem step by step, we will follow the information given in the question and derive the lengths of the two trains A and B, and then find the difference between their lengths.

### Step 1: Define the lengths of the trains
Let the length of train A be \( l_1 \) meters and the length of train B be \( l_2 \) meters. According to the problem, we have:
\[ l_1 + l_2 = 660 \quad \text{(Equation 1)} \]

### Step 2: Define the speeds of the trains
The ratio of the speed of train A to train B is given as 5:8. We can express the speeds as:
\[ \text{Speed of A} = 5k \quad \text{and} \quad \text{Speed of B} = 8k \]
where \( k \) is a common factor.

### Step 3: Define the time taken to cross an electric pole
The time taken for a train to cross an electric pole is given by the formula:
\[ \text{Time} = \frac{\text{Length of Train}}{\text{Speed of Train}} \]
For train A:
\[ \text{Time taken by A} = \frac{l_1}{5k} \]
For train B:
\[ \text{Time taken by B} = \frac{l_2}{8k} \]

### Step 4: Set up the ratio of times
According to the problem, the ratio of the time taken by train A to that taken by train B is 4:3. Thus, we have:
\[ \frac{\frac{l_1}{5k}}{\frac{l_2}{8k}} = \frac{4}{3} \]
This simplifies to:
\[ \frac{8l_1}{5l_2} = \frac{4}{3} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 8l_1 \cdot 3 = 4 \cdot 5l_2 \]
This simplifies to:
\[ 24l_1 = 20l_2 \quad \text{(Equation 2)} \]

### Step 6: Express \( l_1 \) in terms of \( l_2 \)
From Equation 2, we can express \( l_1 \) in terms of \( l_2 \):
\[ l_1 = \frac{20}{24} l_2 = \frac{5}{6} l_2 \]

### Step 7: Substitute \( l_1 \) in Equation 1
Now substitute \( l_1 \) in Equation 1:
\[ \frac{5}{6}l_2 + l_2 = 660 \]
Combining the terms gives:
\[ \frac{5}{6}l_2 + \frac{6}{6}l_2 = 660 \]
\[ \frac{11}{6}l_2 = 660 \]

### Step 8: Solve for \( l_2 \)
Multiplying both sides by \( \frac{6}{11} \):
\[ l_2 = 660 \cdot \frac{6}{11} = 60 \cdot 6 = 360 \text{ meters} \]

### Step 9: Find \( l_1 \)
Now substitute \( l_2 \) back to find \( l_1 \):
\[ l_1 = 660 - l_2 = 660 - 360 = 300 \text{ meters} \]

### Step 10: Calculate the difference between the lengths
The difference between the lengths of the two trains is:
\[ l_2 - l_1 = 360 - 300 = 60 \text{ meters} \]

### Final Answer
The difference between the lengths of the two trains is **60 meters**.

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