Sharmi starts from Pune and Ritesh starts from Mumbai to meet each other. The speed of Sharmi is 37 km/h and the speed of Ritesh is 41 km/h. If the distance between Pune and Mumbai is 234 km and both start their journeys at the same time, after how many hours from the start will they meet?

Introduction / Context:
This is a classic “two people moving towards each other” time and distance problem. Sharmi and Ritesh start from two different cities, Pune and Mumbai, at the same time and travel towards each other along the same straight route. We are given their speeds and the distance between the cities, and we must determine the time after the start at which they will meet. This concept is widely used in aptitude tests to check understanding of relative speed.

Given Data / Assumptions:
- Distance between Pune and Mumbai = 234 km.
- Speed of Sharmi = 37 km/h.
- Speed of Ritesh = 41 km/h.
- Both start at the same time and move directly towards each other along a straight path.
- Their speeds remain constant throughout the motion.

Concept / Approach:
When two objects move directly towards each other on a straight line, their relative speed is the sum of their individual speeds. The total distance between them initially is 234 km, and they close this gap at the rate of (37 + 41) km/h. The time taken to meet is simply the initial distance divided by this combined speed. This method avoids tracking each traveller separately because their combined progress towards reducing the distance is what matters.

Step-by-Step Solution:
Step 1: Compute the relative speed.Relative speed = 37 km/h + 41 km/h = 78 km/h.Step 2: Use the formula time = distance / speed.Time to meet = 234 km / 78 km/h.Step 3: Simplify 234 / 78.Both numerator and denominator are divisible by 78, so 234 / 78 = 3 hours.

Verification / Alternative check:
In 3 hours, Sharmi travels 37 * 3 = 111 km and Ritesh travels 41 * 3 = 123 km. The sum of these distances is 111 + 123 = 234 km, exactly matching the distance between Pune and Mumbai. This means that after 3 hours they will meet somewhere in between the two cities. Since the numbers match perfectly, our result is consistent with the problem conditions.

Why Other Options Are Wrong:
- 2.5 hours would give a total closed distance of 78 * 2.5 = 195 km, which is less than 234 km, so they would not have met yet.
- 4 hours would give 78 * 4 = 312 km, which exceeds the actual distance, so this cannot be the meeting time.
- 3.5 hours would give 78 * 3.5 = 273 km, again exceeding the initial separation.

Common Pitfalls:
Some candidates mistakenly subtract speeds instead of adding them when the objects move towards each other. Others may calculate the distances for each person separately and then struggle to coordinate the time. Remember that when objects approach each other on a straight line, you always add their speeds to get the relative speed. Using this idea keeps the calculation short and reliable.

Final Answer:
They will meet 3 hours after they start their journeys.