Train A crosses a 230 meters long platform in 29 seconds and train B crosses a 150 meters long platform in 24 seconds. Train B having length of 450 meters crosses train A in 160 seconds, while running in the same direction. Find how much time will the train A take to cross a 50 meters long bridge (Speed of train B > speed of train A)?

The correct answer is 20 seconds.

Step 1: Understand the given information and set up variables.

Let $L_A$ be the length of train A in meters.
Let $S_A$ be the speed of train A in m/s.
Let $L_B$ be the length of train B in meters, given as $450\text{ m}$.
Let $S_B$ be the speed of train B in m/s.

Step 2: Formulate equations for Train A and Train B crossing platforms.

When a train crosses a platform, the total distance covered equals the sum of the length of the train and the length of the platform.

For Train A crossing a 230-meter platform in 29 seconds:
$\text{Distance}=L_A+230$

$S_A=\frac{L_A+230}{29}$

$29\×S_A=L_A+230$     — (Equation 1)

For Train B (length 450 m) crossing a 150-meter platform in 24 seconds:
$\text{Distance}=450+150=600\text{ m}$

$S_B=\frac{600}{24}=25\text{ m/s}$

Step 3: Analyze the scenario where Train B crosses Train A.

Since both trains are running in the same direction and speed of Train B > speed of Train A, the relative speed is $S_B-S_A=25-S_A$.

The total distance covered while crossing each other is the sum of their lengths: $L_A+L_B=L_A+450$.

The time taken is 160 seconds:

$\frac{L_A+450}{25-S_A}=160$

$L_A+450=160\×(25-S_A)$

$L_A+450=4000-160\×S_A$

$L_A+160\×S_A=3550$     — (Equation 2)

Step 4: Solve for Train A's length and speed.

From Equation 1, substitute $L_A=29\×S_A-230$ into Equation 2:

$(29\×S_A-230)+160\×S_A=3550$

$189\×S_A-230=3550$

$189\×S_A=3780$

$S_A=\frac{3780}{189}=20\text{ m/s}$

Now calculate the length of Train A ($L_A$):

$L_A=29\×20-230=580-230=350\text{ m}$

Step 5: Calculate time taken for Train A to cross a 50-meter long bridge.

Total distance to cross the bridge = Length of Train A + Length of Bridge

$\text{Total Distance}=350+50=400\text{ m}$

Time taken = $\frac{\text{Total Distance}}{S_A}$

$\text{Time}=\frac{400}{20}=20\text{ seconds}$