Ramu rides his bike at an average speed of 45 km/h and reaches his destination in 4 hours. Somu covers the same distance in 6 hours. If Ramu rides at a speed that is 9 km/h less than 45 km/h and Somu rides at a speed that is 10 km/h more than 30 km/h, what is the difference in their times taken to reach the destination, in minutes?

Introduction / Context:
This question tests understanding of average speed, total distance, and how changes in speed affect travel time. Two people travel the same distance under different conditions, and the question asks for the difference in their new travel times. Such problems help students practice using the fundamental formula time = distance / speed.

Given Data / Assumptions:

• Ramu's original speed = 45 km/h.
• Ramu's travel time = 4 hours.
• Somu's travel time for the same distance = 6 hours.
• Somu's original speed = distance / time.
• New speed of Ramu = 45 - 9 = 36 km/h.
• New speed of Somu = his original speed + 10 km/h.
• We must find the difference between the new travel times in minutes.

Concept / Approach:
The distance between the two locations is the same for both riders, so we can calculate it once using any one rider's initial data. After we know the distance, we can compute new times at the changed speeds using time = distance / speed. The required difference in time is then converted from hours to minutes by multiplying by 60.

Step-by-Step Solution:
Distance D = Ramu's speed * time = 45 * 4 = 180 km. Somu's original speed = distance / time = 180 / 6 = 30 km/h. New speed of Ramu = 45 - 9 = 36 km/h. New speed of Somu = 30 + 10 = 40 km/h. New time for Ramu = 180 / 36 = 5 hours. New time for Somu = 180 / 40 = 4.5 hours. Difference in time = 5 - 4.5 = 0.5 hours. Convert to minutes: 0.5 * 60 = 30 minutes.

Verification / Alternative check:
We can quickly verify the logic by checking if the new speeds are reasonable. Ramu slows down from 45 km/h to 36 km/h, so his travel time should increase from 4 hours to 5 hours. Somu speeds up from 30 km/h to 40 km/h, so his travel time should reduce from 6 hours to 4.5 hours. The difference between 5 hours and 4.5 hours is clearly 0.5 hours or 30 minutes, confirming our calculation.

Why Other Options Are Wrong:
18 minutes, 27 minutes, 32 minutes, and 24 minutes do not correspond to the actual difference in travel times obtained from the correct distance and speed calculations. These values may result from arithmetic mistakes, such as incorrect computation of new speeds or using the original times instead of the new ones. Only 30 minutes matches the correct time difference.

Common Pitfalls:
Students sometimes forget to compute Somu's initial speed correctly and instead guess it. Another mistake is to modify times instead of speeds, or to subtract the original times rather than the new ones. A clear step-by-step calculation starting with the common distance and then finding each new time helps prevent these errors.

Final Answer:
The difference in their times to reach the destination under the new speeds is 30 minutes.