Abhinav leaves Mumbai at 6 am and reaches Bangalore at 10 am Praveen leaves Bangalore at 8 am and reaches Mumbai at 11 : 30 am. At what time do they cross each other?

Options

A. `10` am
B. `8:32` am
C. `8:56` am
D. `9:20` am

Answer

To solve the problem step by step, let's break down the information given and calculate the time when Abhinav and Praveen cross each other.

### Step 1: Determine the total distance between Mumbai and Bangalore
- Abhinav leaves Mumbai at 6 AM and reaches Bangalore at 10 AM, taking a total of 4 hours.
- Praveen leaves Bangalore at 8 AM and reaches Mumbai at 11:30 AM, taking a total of 3.5 hours (or 7/2 hours).

### Step 2: Calculate their speeds
- Let the total distance between Mumbai and Bangalore be \( D \).
- Speed of Abhinav = \( \frac{D}{4} \)
- Speed of Praveen = \( \frac{D}{7/2} = \frac{2D}{7} \)

### Step 3: Find a common distance for calculation
- To simplify calculations, we can assume \( D = 28 \) km (the least common multiple of 4 and 7/2).
- Speed of Abhinav = \( \frac{28}{4} = 7 \) km/h
- Speed of Praveen = \( \frac{28}{7/2} = 8 \) km/h

### Step 4: Calculate the distance covered by Abhinav before Praveen starts
- From 6 AM to 8 AM, Abhinav travels for 2 hours.
- Distance covered by Abhinav in 2 hours = Speed × Time = \( 7 \text{ km/h} \times 2 \text{ h} = 14 \text{ km} \).

### Step 5: Calculate the remaining distance
- Total distance = 28 km
- Distance covered by Abhinav = 14 km
- Remaining distance = \( 28 - 14 = 14 \) km.

### Step 6: Determine the time taken for them to meet after Praveen starts
- After 8 AM, both are moving towards each other.
- Their combined speed = Speed of Abhinav + Speed of Praveen = \( 7 + 8 = 15 \) km/h.
- Time taken to cover the remaining distance = \( \frac{\text{Distance}}{\text{Relative Speed}} = \frac{14 \text{ km}}{15 \text{ km/h}} = \frac{14}{15} \text{ hours} \).

### Step 7: Convert time to minutes
- \( \frac{14}{15} \text{ hours} = 0.9333 \text{ hours} \times 60 \text{ minutes/hour} = 56 \text{ minutes} \).

### Step 8: Calculate the crossing time
- Praveen starts at 8 AM.
- They will meet \( 56 \) minutes after 8 AM.
- Therefore, they cross each other at \( 8:56 \) AM.

### Final Answer
Abhinav and Praveen cross each other at **8:56 AM**.

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