Two trains take `3s` to pass another when going in the opposite directions but only `2.5s` if the speed of one is increased by `50%`. The time one would take to pass the other when going in the same direction at their original speed is

Options

A. `10s`
B. `12s`
C. `15s`
D. `18s`

Answer

To solve the problem, we need to analyze the information given about the two trains and their speeds. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( V_1 \) = speed of the first train
- \( V_2 \) = speed of the second train
- \( L_1 \) = length of the first train
- \( L_2 \) = length of the second train

### Step 2: Write the Equation for Opposite Directions
When the two trains pass each other in opposite directions, the time taken is given as \( 3 \) seconds. The relative speed when they are moving towards each other is \( V_1 + V_2 \). The total distance covered when they pass each other is \( L_1 + L_2 \).

Using the formula for time, we have:
\[ 3 = \frac{L_1 + L_2}{V_1 + V_2} \]
From this, we can derive:
\[ L_1 + L_2 = 3(V_1 + V_2) \quad \text{(Equation 1)} \]

### Step 3: Write the Equation for Increased Speed
When the speed of the first train is increased by \( 50\% \), its new speed becomes \( 1.5 V_1 \). The time taken to pass each other in this case is \( 2.5 \) seconds. The relative speed now is \( 1.5 V_1 + V_2 \).

Using the time formula again, we have:
\[ 2.5 = \frac{L_1 + L_2}{1.5 V_1 + V_2} \]
From this, we can derive:
\[ L_1 + L_2 = 2.5(1.5 V_1 + V_2) \quad \text{(Equation 2)} \]

### Step 4: Set the Two Equations Equal
Since both equations equal \( L_1 + L_2 \), we can set them equal to each other:
\[ 3(V_1 + V_2) = 2.5(1.5 V_1 + V_2) \]

### Step 5: Expand and Simplify
Expanding both sides gives:
\[ 3V_1 + 3V_2 = 3.75V_1 + 2.5V_2 \]
Rearranging terms:
\[ 3V_1 + 3V_2 - 3.75V_1 - 2.5V_2 = 0 \]
This simplifies to:
\[ -0.75V_1 + 0.5V_2 = 0 \]
Thus:
\[ 0.5V_2 = 0.75V_1 \quad \Rightarrow \quad V_2 = \frac{1.5}{1} V_1 \]

### Step 6: Substitute Back to Find Time in Same Direction
Now we know that \( V_2 = 1.5 V_1 \). We want to find the time taken when both trains are moving in the same direction. The relative speed in this case is:
\[ V_2 - V_1 = 1.5V_1 - V_1 = 0.5V_1 \]

Using the equation for time:
\[ \text{Time} = \frac{L_1 + L_2}{V_2 - V_1} \]
Substituting \( L_1 + L_2 \) from Equation 1:
\[ \text{Time} = \frac{3(V_1 + V_2)}{0.5V_1} \]
Substituting \( V_2 = 1.5V_1 \):
\[ \text{Time} = \frac{3(V_1 + 1.5V_1)}{0.5V_1} = \frac{3(2.5V_1)}{0.5V_1} = \frac{7.5V_1}{0.5V_1} = 15 \text{ seconds} \]

### Final Answer
The time taken for one train to pass the other when going in the same direction at their original speeds is **15 seconds**.

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