Introduction / Context:
This question checks your ability to convert between time and speed for a fixed distance. When the distance remains constant, average speed is inversely proportional to time taken. The problem asks for the increase in speed required to reduce the lap time from 15 minutes to 12 minutes for a racing car.
Given Data / Assumptions:
- Initial average speed of the car = 108 km/h.
- Initial time for one lap = 15 minutes.
- Required time for one lap = 12 minutes.
- The lap distance is fixed and the car runs at constant average speeds.
Concept / Approach:
First find the actual distance of one lap using the original speed and time. Once the lap distance is known, use the target time and the same distance to compute the new required speed. The difference between new and old speeds gives the increase in speed.
Step-by-Step Solution:
Convert 15 minutes to hours: 15 / 60 = 0.25 hours.
Lap distance D = speed * time = 108 * 0.25 = 27 km.
Now convert 12 minutes to hours: 12 / 60 = 0.2 hours.
Required new speed v_new = D / time = 27 / 0.2 km/h.
Compute v_new = 27 / 0.2 = 135 km/h.
Increase in speed = v_new - original speed = 135 - 108 = 27 km/h.
Verification / Alternative check:
At 135 km/h for 0.2 hours, distance = 135 * 0.2 = 27 km.
At 108 km/h for 0.25 hours, distance = 108 * 0.25 = 27 km.
Both distances are equal, confirming that the same lap is completed faster only because of higher speed.
Why Other Options Are Wrong:
24 km/h increase would give new speed 132 km/h, distance in 12 minutes = 132 * 0.2 = 26.4 km, not equal to 27 km.
21 km/h increase leads to 129 km/h and distance 25.8 km in 12 minutes, which is short.
30 km/h increase leads to 138 km/h and distance 27.6 km, which is longer than the actual lap.
Common Pitfalls:
Some learners directly take a ratio of times and incorrectly subtract percentages without converting to actual distances.
Another common mistake is to keep times in minutes while using speeds in km/h without converting to hours.
Always ensure units are consistent and compute the real distance once, then reuse it for the new scenario.
Final Answer:
The car must increase its speed by 27 km/h.