The distance between Lucknow and Delhi is 700 km. Rajdhani express starts from Delhi for the Lucknow at 60 km/h. 50 minutes later Lucknow express leaves Lucknow for Delhi on the parallel tracks at 70 km/h. How far from Lucknow will they cross each other?

Options

A. 250 km
B. 360 km
C. 350 km
D. 475 km

Answer

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

### Step 1: Understand the problem
We have two trains: Rajdhani Express starting from Delhi to Lucknow at a speed of 60 km/h, and Lucknow Express starting from Lucknow to Delhi at a speed of 70 km/h. The Lucknow Express starts 50 minutes after the Rajdhani Express.

### Step 2: Convert time into hours
The first step is to convert the 50 minutes into hours since the speeds are given in km/h.
\[ 50 \text{ minutes} = \frac{50}{60} \text{ hours} = \frac{5}{6} \text{ hours} \]

### Step 3: Calculate the distance covered by Rajdhani Express in 50 minutes
Now, we calculate how far the Rajdhani Express travels in that time.
\[ \text{Distance} = \text{Speed} \times \text{Time} = 60 \text{ km/h} \times \frac{5}{6} \text{ hours} = 50 \text{ km} \]

### Step 4: Determine the remaining distance
Since the total distance between Lucknow and Delhi is 700 km, we can find the remaining distance after the Rajdhani Express has traveled 50 km.
\[ \text{Remaining distance} = 700 \text{ km} - 50 \text{ km} = 650 \text{ km} \]

### Step 5: Calculate the relative speed
Both trains are moving towards each other, so we can add their speeds to find the relative speed.
\[ \text{Relative Speed} = 60 \text{ km/h} + 70 \text{ km/h} = 130 \text{ km/h} \]

### Step 6: Calculate the time taken to meet
Now, we can find the time it takes for the two trains to meet using the remaining distance and the relative speed.
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{650 \text{ km}}{130 \text{ km/h}} = 5 \text{ hours} \]

### Step 7: Calculate the distance covered by Lucknow Express
Now we need to find out how far the Lucknow Express travels in those 5 hours.
\[ \text{Distance} = \text{Speed} \times \text{Time} = 70 \text{ km/h} \times 5 \text{ hours} = 350 \text{ km} \]

### Conclusion
Thus, the distance from Lucknow where the two trains cross each other is:
\[ \text{Distance from Lucknow} = 350 \text{ km} \]

### Final Answer
The trains will cross each other 350 km from Lucknow.

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