Amarnath express left Amritsar for Gorakhpur. Two hours later Gorakhnath express left from Amritsar to Gorakhpur. Both trains reached Gorakhpur simultaneously. If Amarnath express had started from Amritsar and Gorakhnath express had started from Gorakhpur at the same time and travelled towards each other they would meet in 1 h 20 min. Find the time taken by Amarnath express to travel from Amritsar to Gorakhpur (in hours) :

Options

A. 2
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to establish the relationship between the speeds of the two trains and the time taken by each train to travel from their respective starting points to Gorakhpur.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the time taken by Amarnath Express to travel from Amritsar to Gorakhpur be \( h \) hours.
- Therefore, the time taken by Gorakhnath Express to travel from Gorakhpur to Amritsar will be \( h - 2 \) hours (since it leaves 2 hours later).

2. **Understanding the Meeting Condition:**
- If both trains started at the same time and traveled towards each other, they would meet in 1 hour and 20 minutes, which is \( \frac{4}{3} \) hours.

3. **Setting Up the Equation:**
- The speeds of the trains can be expressed as:
- Speed of Amarnath Express = \( \frac{D}{h} \)
- Speed of Gorakhnath Express = \( \frac{D}{h - 2} \)
- When they travel towards each other, the sum of their speeds equals the distance covered in \( \frac{4}{3} \) hours:
\[ \frac{D}{h} + \frac{D}{h - 2} = \frac{D}{\frac{4}{3}} \]
- Simplifying gives:
\[ \frac{D}{h} + \frac{D}{h - 2} = \frac{3D}{4} \]

4. **Canceling D:**
- Since \( D \) (the distance) is common on both sides, we can cancel it out:
\[ \frac{1}{h} + \frac{1}{h - 2} = \frac{3}{4} \]

5. **Finding a Common Denominator:**
- The common denominator for the left side is \( h(h - 2) \):
\[ \frac{(h - 2) + h}{h(h - 2)} = \frac{3}{4} \]
- This simplifies to:
\[ \frac{2h - 2}{h(h - 2)} = \frac{3}{4} \]

6. **Cross Multiplying:**
- Cross-multiplying gives:
\[ 4(2h - 2) = 3h(h - 2) \]
- Expanding both sides:
\[ 8h - 8 = 3h^2 - 6h \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ 3h^2 - 14h + 8 = 0 \]

8. **Using the Quadratic Formula:**
- We can solve this quadratic equation using the quadratic formula \( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 3 \), \( b = -14 \), and \( c = 8 \):
\[ h = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 3 \cdot 8}}{2 \cdot 3} \]
\[ h = \frac{14 \pm \sqrt{196 - 96}}{6} \]
\[ h = \frac{14 \pm \sqrt{100}}{6} \]
\[ h = \frac{14 \pm 10}{6} \]

9. **Calculating Possible Values for h:**
- This gives us two possible values:
\[ h = \frac{24}{6} = 4 \quad \text{and} \quad h = \frac{4}{6} = \frac{2}{3} \]
- Since time cannot be negative or less than the time taken by Gorakhnath Express, we take \( h = 4 \).

### Final Answer:
The time taken by Amarnath Express to travel from Amritsar to Gorakhpur is **4 hours**.