Train-A leaves Delhi at 8:00 A.M. and reaches Lucknow at 4:00 P.M. and train-B leaves Lucknow at 1:00 P.M. and reaches Delhi at 5:00 P.M. Find at what time both trains will meet each other?

Options

A. A)`2:30` P.M.
B. B)`2:00` P.M.
C. C)`3:00` P.M.
D. D)`2:15` P.M.

Answer

To solve the problem of when Train A and Train B will meet, we can follow these steps:

### Step 1: Determine the travel times of both trains.
- Train A leaves Delhi at 8:00 A.M. and reaches Lucknow at 4:00 P.M.
- Total travel time for Train A = 4:00 P.M. - 8:00 A.M. = 8 hours.

- Train B leaves Lucknow at 1:00 P.M. and reaches Delhi at 5:00 P.M.
- Total travel time for Train B = 5:00 P.M. - 1:00 P.M. = 4 hours.

### Step 2: Calculate the speeds of both trains.
- Let the distance between Delhi and Lucknow be D kilometers.
- Speed of Train A = Distance / Time = D / 8 km/h.
- Speed of Train B = Distance / Time = D / 4 km/h.

### Step 3: Determine the distance covered by each train until they meet.
- Train A starts at 8:00 A.M. and travels until the meeting time.
- Train B starts at 1:00 P.M. and travels until the meeting time.

Let the meeting time be T hours after 8:00 A.M.
- Distance covered by Train A by time T = Speed × Time = (D/8) × T.
- Distance covered by Train B by time T = Speed × Time = (D/4) × (T - 5) (since Train B starts 5 hours later).

### Step 4: Set up the equation for the distances.
Since both trains are traveling towards each other, the sum of the distances they cover will equal the total distance D:
\[ \frac{D}{8} \times T + \frac{D}{4} \times (T - 5) = D \]

### Step 5: Simplify the equation.
1. Multiply through by 8 to eliminate the denominators:
\[ D \times T + 2D \times (T - 5) = 8D \]
2. Distributing the terms:
\[ D \times T + 2D \times T - 10D = 8D \]
3. Combine like terms:
\[ 3D \times T - 10D = 8D \]
4. Add 10D to both sides:
\[ 3D \times T = 18D \]
5. Divide both sides by 3D (assuming D ≠ 0):
\[ T = 6 \text{ hours} \]

### Step 6: Calculate the meeting time.
- Since T = 6 hours after 8:00 A.M., the meeting time is:
\[ 8:00 \text{ A.M.} + 6 \text{ hours} = 2:00 \text{ P.M.} \]

### Final Answer:
Both trains will meet each other at **2:00 P.M.**.

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