Two trains A and B crosses each other while running in opposite direction in half of a minute, If train A running with double of its earlier speed and train B reduces its speed to 50% of its earlier speed, then they again crosses each other in the same time while running in opposite direction. Find the length of both train together, if speed of train - A is 36 kmph. (in m)

Options

A. A)500
B. B)600
C. C)400
D. D)900

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Given Information
- Train A's initial speed = 36 km/h
- They cross each other in 30 seconds (half a minute).
- Train A doubles its speed.
- Train B reduces its speed to 50% of its earlier speed.

### Step 2: Convert Train A's Speed to m/s
To work with the time in seconds, we need to convert the speed from km/h to m/s:
\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]
For Train A:
\[ \text{Speed of A} = 36 \times \frac{5}{18} = 10 \text{ m/s} \]

### Step 3: Set Up the Equation for the First Crossing
Let the speed of Train B be \( V_B \) km/h. When both trains cross each other, the total distance (length of Train A + length of Train B) can be expressed as:
\[ \text{Distance} = (\text{Speed of A} + \text{Speed of B}) \times \text{Time} \]
In terms of m/s, this becomes:
\[ \text{Distance} = (10 + V_B \times \frac{5}{18}) \times 30 \]

### Step 4: Set Up the Equation for the Second Crossing
After the speed changes:
- Speed of Train A becomes \( 2 \times 36 = 72 \) km/h (which is \( 72 \times \frac{5}{18} = 20 \) m/s).
- Speed of Train B becomes \( \frac{V_B}{2} \) km/h (which is \( \frac{V_B}{2} \times \frac{5}{18} \) m/s).

The distance for the second crossing is:
\[ \text{Distance} = (20 + \frac{V_B}{2} \times \frac{5}{18}) \times 30 \]

### Step 5: Equate the Two Distances
Since the distances are equal:
\[ (10 + V_B \times \frac{5}{18}) \times 30 = (20 + \frac{V_B}{2} \times \frac{5}{18}) \times 30 \]
Dividing both sides by 30:
\[ 10 + V_B \times \frac{5}{18} = 20 + \frac{V_B}{2} \times \frac{5}{18} \]

### Step 6: Solve for \( V_B \)
Rearranging the equation:
\[ V_B \times \frac{5}{18} - \frac{V_B}{2} \times \frac{5}{18} = 20 - 10 \]
Factoring out \( \frac{5}{18} \):
\[ \frac{5}{18} \left( V_B - \frac{V_B}{2} \right) = 10 \]
This simplifies to:
\[ \frac{5}{18} \left( \frac{V_B}{2} \right) = 10 \]
Multiplying both sides by \( \frac{18}{5} \):
\[ \frac{V_B}{2} = 36 \implies V_B = 72 \text{ km/h} \]

### Step 7: Calculate the Total Length of Both Trains
Now we can find the total length of both trains:
\[ \text{Total Length} = (10 + 20) \times 30 = 30 \times 30 = 900 \text{ meters} \]

### Final Answer
The total length of both trains together is **900 meters**.

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