Person K cycles from point A to point B at 10 km/h and returns from B to A at 9 km/h. Person L cycles both ways at 12 km/h. Over the complete trip, L takes 10 minutes less than K. What is the distance (in km) between A and B?

Introduction / Context:
This is a classic average speed and time comparison problem where two people travel the same route with different speeds. By comparing total times for round trips, we can find the distance between two points. The problem focuses on forming time equations from distance and speed, then using the given time difference to solve for the distance.

Given Data / Assumptions:

• K goes from A to B at 10 km/h and from B to A at 9 km/h.
• L goes from A to B and back at 12 km/h in both directions.
• L takes 10 minutes (that is 10 / 60 = 1 / 6 hour) less than K for the round trip.
• Both travel the same distance between A and B.

Concept / Approach:
Let the one way distance be d km. Then the total time for K is d / 10 + d / 9 hours. The total time for L is 2d / 12 hours. The difference between these total times is 1 / 6 hour. Setting up this equation and solving for d gives the required distance.

Step-by-Step Solution:
Let distance A to B be d km. Time taken by K = d / 10 + d / 9 = (9d + 10d) / 90 = 19d / 90 hours. Time taken by L = 2d / 12 = d / 6 hours. Given that K takes 1 / 6 hour more: 19d / 90 - d / 6 = 1 / 6. Convert d / 6 to /90: d / 6 = 15d / 90. So 19d / 90 - 15d / 90 = 1 / 6. This gives 4d / 90 = 1 / 6. Simplify: 2d / 45 = 1 / 6 so 2d = 45 / 6. Therefore, d = 45 / 12 = 15 / 4 = 3.75 km.

Verification / Alternative check:
Time taken by K = 19 * 3.75 / 90 hours = 0.7917 hours (approximately 47.5 minutes). Time taken by L = 3.75 / 6 = 0.625 hours (37.5 minutes). The difference is about 10 minutes, which matches the condition. So the distance is correct.

Why Other Options Are Wrong:

• 3.25 km, 4.00 km and 4.50 km all give time differences that are not equal to 10 minutes when substituted into the time equations.

Common Pitfalls:
Candidates sometimes average speeds instead of using time formulas, which is incorrect for round trips. Another common error is forgetting to convert 10 minutes to hours or mishandling fractional arithmetic, leading to an incorrect distance.

Final Answer:
The distance between points A and B is 3.75 km.