Two trains Katrina Express and Madhuri express cross an electric pole in 32 sec. and 40 sec. respectively. Length of Madhuri Express is `37.5%` less than Katrina express. Then in how much time they will cross each other if they are moving in opposite direction.

Options

A. `34.6` sec
B. `38.2` sec
C. `35.4` sec
D. `32.8` sec

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the Lengths of the Trains
Let the length of Katrina Express be \( L_k \) meters. Since the length of Madhuri Express is 37.5% less than that of Katrina Express, we can express the length of Madhuri Express as:
\[ L_m = L_k - 0.375L_k = 0.625L_k \]

### Step 2: Calculate the Speeds of the Trains
The speed of a train can be calculated using the formula:
\[ \text{Speed} = \frac{\text{Length}}{\text{Time}} \]

For Katrina Express, which crosses an electric pole in 32 seconds:
\[ \text{Speed of Katrina Express} = \frac{L_k}{32} \]

For Madhuri Express, which crosses an electric pole in 40 seconds:
\[ \text{Speed of Madhuri Express} = \frac{L_m}{40} = \frac{0.625L_k}{40} \]

### Step 3: Simplify the Speeds
Now, we can simplify the speed of Madhuri Express:
\[ \text{Speed of Madhuri Express} = \frac{0.625L_k}{40} = \frac{L_k}{64} \]

### Step 4: Calculate the Combined Speed
When two trains move in opposite directions, their speeds add up. Therefore, the combined speed is:
\[ \text{Combined Speed} = \text{Speed of Katrina Express} + \text{Speed of Madhuri Express} \]
\[ = \frac{L_k}{32} + \frac{L_k}{64} \]

To add these fractions, we need a common denominator, which is 64:
\[ \frac{L_k}{32} = \frac{2L_k}{64} \]
Thus,
\[ \text{Combined Speed} = \frac{2L_k}{64} + \frac{L_k}{64} = \frac{3L_k}{64} \]

### Step 5: Calculate the Total Length to be Crossed
The total length to be crossed when the two trains meet is the sum of their lengths:
\[ \text{Total Length} = L_k + L_m = L_k + 0.625L_k = 1.625L_k \]

### Step 6: Calculate the Time Taken to Cross Each Other
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
We can substitute the total length and the combined speed:
\[ \text{Time} = \frac{1.625L_k}{\frac{3L_k}{64}} = \frac{1.625L_k \times 64}{3L_k} \]
The \( L_k \) cancels out:
\[ \text{Time} = \frac{1.625 \times 64}{3} \]

### Step 7: Calculate the Final Time
Calculating \( 1.625 \times 64 \):
\[ 1.625 \times 64 = 104 \]
Thus,
\[ \text{Time} = \frac{104}{3} \approx 34.67 \text{ seconds} \]

### Final Answer
The two trains will cross each other in approximately \( 34.67 \) seconds.

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