Akash leaves Mumbai at 6 a.m. and reaches Bangalore at 10 a.m. Prakash leaves Bangalore at 8 a.m. and reaches Mumbai at 11:30 a.m. Assuming they travel at constant speeds along the same route, at what time do Akash and Prakash meet each other?

Introduction / Context:
This is another relative speed problem, but instead of knowing the distance between the two cities directly, you are given the times that each person takes to travel the entire route. Akash and Prakash start at different times and travel at different constant speeds. The question asks at what time they cross each other on the route.

Given Data / Assumptions:

• Akash travels from Mumbai to Bangalore, leaving at 6 a.m. and reaching at 10 a.m., so his total travel time is 4 hours.
• Prakash travels from Bangalore to Mumbai, leaving at 8 a.m. and reaching at 11:30 a.m., so his total travel time is 3.5 hours.
• They travel along the same straight route with constant speeds.
• We must find the time at which they meet.

Concept / Approach:
Let the distance between Mumbai and Bangalore be D km. Then Akash’s speed is D / 4 km/h and Prakash’s speed is D / 3.5 km/h. Let t hours be the time after 6 a.m. when they meet. At that time, Akash has been travelling for t hours, while Prakash has been travelling for t - 2 hours because he starts at 8 a.m. Their total distance covered at the meeting moment equals D.

Step-by-Step Solution:
Step 1: Speed of Akash v_A = D / 4 km/h. Step 2: Speed of Prakash v_P = D / 3.5 km/h. Step 3: Let t be the time in hours after 6 a.m. when they meet. Then Akash has travelled t hours, Prakash has travelled t - 2 hours. Step 4: Distance covered by Akash = v_A * t = (D / 4) * t. Step 5: Distance covered by Prakash = v_P * (t - 2) = (D / 3.5) * (t - 2). Step 6: At meeting, sum of distances = D, so (D / 4) * t + (D / 3.5) * (t - 2) = D. Step 7: Divide entire equation by D to get t / 4 + (t - 2) / 3.5 = 1. Step 8: Solve: t / 4 + (t - 2) / 3.5 = 1 gives t = 44 / 15 hours. Step 9: 44 / 15 hours ≈ 2.933 hours, which is 2 hours 56 minutes after 6 a.m., so meeting time is 8:56 a.m.
Verification / Alternative check:
At 8:56 a.m., Akash has travelled 2 hours 56 minutes, and Prakash has travelled 56 minutes. Akash’s speed is D / 4; distance covered by him is (D / 4) * (44 / 15) = 11D / 15. Prakash’s speed is D / 3.5 = 2D / 7; distance covered by him in 44 / 15 - 2 = 14 / 15 hours is (2D / 7) * (14 / 15) = 4D / 15. Total distance equals 11D / 15 + 4D / 15 = D, confirming that they meet at this time.

Why Other Options Are Wrong:

• 10 a.m.: Too late; by this time they would already have crossed each other.
• 8:32 a.m.: Too early; the algebraic equation does not support this time.
• 9:20 a.m.: Also inconsistent with the ratio of their speeds and travel times.

Common Pitfalls:

• Forgetting that Prakash starts 2 hours after Akash, leading to incorrect time expressions.
• Cancelling distance D incorrectly and making algebraic mistakes when solving for t.
• Converting fractional hours to minutes inaccurately.

Final Answer:
Akash and Prakash meet each other at 8:56 a.m.