Pushpak Express leaves Lucknow at 6 a.m. and 2 hours later another train Bhopal Express leaves Lucknow both train arrive Bhopal at 4 P.M on the same day with the difference between their speeds be 10 km per hour what is the average speeds of both the trains over the entire route

Options

A. 40 km/h
B. 44(4/9)km/h
C. 42(3/5)km/h
D. none of these

Answer

To solve the problem, we need to determine the average speeds of both trains, Pushpak Express and Bhopal Express, based on the information provided.

### Step-by-Step Solution:

1. **Identify the Departure and Arrival Times:**
- Pushpak Express leaves Lucknow at 6 a.m.
- Bhopal Express leaves Lucknow at 8 a.m. (2 hours later).
- Both trains arrive in Bhopal at 4 p.m.

2. **Calculate the Travel Time for Each Train:**
- Pushpak Express travels from 6 a.m. to 4 p.m., which is 10 hours.
- Bhopal Express travels from 8 a.m. to 4 p.m., which is 8 hours.

3. **Let the Speed of Pushpak Express be \( x \) km/h:**
- Therefore, the speed of Bhopal Express will be \( x + 10 \) km/h (since the difference in their speeds is 10 km/h).

4. **Set Up the Distance Equation:**
- The distance traveled by Pushpak Express can be expressed as:
\[ \text{Distance} = \text{Speed} \times \text{Time} = x \times 10 \]
- The distance traveled by Bhopal Express can be expressed as:
\[ \text{Distance} = (x + 10) \times 8 \]

5. **Equate the Distances:**
- Since both trains travel the same distance to Bhopal, we can set the two distance equations equal to each other:
\[ 10x = 8(x + 10) \]

6. **Expand and Simplify the Equation:**
- Expanding the right side:
\[ 10x = 8x + 80 \]
- Rearranging gives:
\[ 10x - 8x = 80 \]
\[ 2x = 80 \]
\[ x = 40 \]

7. **Calculate the Speeds of Both Trains:**
- Speed of Pushpak Express \( = x = 40 \) km/h.
- Speed of Bhopal Express \( = x + 10 = 40 + 10 = 50 \) km/h.

8. **Final Answer:**
- The average speed of Pushpak Express is 40 km/h.
- The average speed of Bhopal Express is 50 km/h.