After repairing, a scooter runs at a speed of 54 km/h, and before repairing it runs at a speed of 48 km/h. It covers a certain distance in 6 hours after repairing. How much time will it take to cover the same distance before repairing?

Introduction / Context:
This is a speed and time comparison problem. The scooter’s speed increases after repair, and we know how long it takes to travel a certain fixed distance at the higher speed. We are asked how long it would take to travel the same distance at the lower pre-repair speed. This type of question reinforces the relationship between speed, distance, and time and the idea that for a fixed distance, time is inversely proportional to speed.

Given Data / Assumptions:
- Speed after repairing = 54 km/h.
- Speed before repairing = 48 km/h.
- Time taken after repairing to cover a certain distance = 6 hours.
- Let the fixed distance be D km.
- The road and driving conditions are assumed to be the same in both cases, with constant speeds over the journey.

Concept / Approach:
First, we determine the actual distance D covered at 54 km/h in 6 hours using D = speed * time. Then, using this same distance and the lower speed of 48 km/h, we find the required time using time = distance / speed. Finally, we convert the resulting decimal hours into hours and minutes to match the answer options. Because the distance is identical in both scenarios, any changes in time are purely due to the difference in speed.

Step-by-Step Solution:
Step 1: Find the distance D using the post-repair speed.D = 54 km/h * 6 h = 324 km.Step 2: Compute the time needed at the pre-repair speed (48 km/h).Time_before = D / 48 = 324 / 48 hours.Step 3: Simplify 324 / 48.Divide numerator and denominator by 12: 324 / 48 = 27 / 4 hours.Step 4: Convert 27 / 4 hours to hours and minutes.27 / 4 = 6.75 hours = 6 hours + 0.75 hours.0.75 hours = 0.75 * 60 = 45 minutes.So time_before = 6 hours 45 minutes.

Verification / Alternative check:
Check consistency by reversing the logic. If the scooter travels 324 km in 6 hours, the speed is 324 / 6 = 54 km/h, which matches the given after-repair speed. At 48 km/h, over 6 hours 45 minutes (which is 6.75 hours), the distance covered is 48 * 6.75 = 324 km, which again matches. Both directions confirm that the computations are correct.

Why Other Options Are Wrong:
- 6 hours 15 minutes corresponds to 6.25 hours, giving a distance of 48 * 6.25 = 300 km, too short.
- 7 hours corresponds to 48 * 7 = 336 km, more than 324 km, so it does not represent the same distance.
- 7 hours 30 minutes is 7.5 hours, giving 48 * 7.5 = 360 km, again larger than the distance covered after repair.

Common Pitfalls:
A common mistake is to try to scale the time directly in proportion to the ratio of speeds without checking the exact arithmetic. While time does scale inversely with speed for a fixed distance, working through the distance first and then recomputing time usually reduces errors. Another pitfall is mishandling decimal hours when converting to minutes; always multiply the fractional part by 60 to get minutes.

Final Answer:
The scooter will take 6 hours 45 minutes to cover the same distance before repairing.