A girl cycles from her home to a certain place. If she cycles at 10 km/h, she arrives at that place at 1 p.m. If she cycles at 15 km/h, she arrives at the same place at 11 a.m. At what constant speed must she cycle so that she reaches there exactly at 12 noon?

Introduction / Context:
This problem explores how travel time changes with speed over a fixed distance. The girl arrives at different times depending on her cycling speed. From these arrival times, we can deduce the distance and starting time, then find the speed required to reach the destination at noon.

Given Data / Assumptions:

• Speed option 1: 10 km/h leads to arrival at 1 p.m.
• Speed option 2: 15 km/h leads to arrival at 11 a.m.
• Distance between home and destination is fixed.
• She always starts at the same time from home.
• We need the speed that results in arrival at 12 noon.

Concept / Approach:
Let the distance be D km and the starting time be T_start. The travel time depends on speed: time = distance / speed. Using two scenarios we can form two equations and solve for D and T_start. Once D is known, we compute the speed needed for the girl to cover D in the time from T_start to 12 noon.

Step-by-Step Solution:
Let distance be D km and starting time be T_start in hours. At 10 km/h, travel time is D / 10 hours, and arrival time is 1 p.m. So T_start + D / 10 = 13:00 hours. At 15 km/h, travel time is D / 15 hours, and arrival time is 11 a.m. So T_start + D / 15 = 11:00 hours. Subtract the second equation from the first: (D / 10) - (D / 15) = 13 - 11 = 2 hours. Compute D / 10 - D / 15 = (3D - 2D) / 30 = D / 30. So D / 30 = 2, hence D = 60 km. Now find T_start from T_start + D / 10 = 13. We have D / 10 = 60 / 10 = 6 hours, so T_start = 13 - 6 = 7 a.m. To arrive at 12 noon, travel time must be from 7 a.m. to 12 p.m., which is 5 hours. Required speed = distance / time = 60 km / 5 h = 12 km/h.

Verification / Alternative check:
Check the original conditions with D = 60 km and T_start = 7 a.m. At 10 km/h, time = 6 hours, arrival at 1 p.m. At 15 km/h, time = 4 hours, arrival at 11 a.m. Both match the given information. For 12 km/h, time = 5 hours, exactly giving arrival at noon, which confirms the solution.

Why Other Options Are Wrong:
Speeds 11, 13 or 14 km/h result in travel times of approximately 5.45, 4.62 or 4.29 hours, leading to arrival times that differ from noon. Similarly, 10 km/h corresponds to a 6 hour trip and arrival at 1 p.m., which does not satisfy the requirement. Only 12 km/h gives the correct noon arrival.

Common Pitfalls:
A common error is to try to average the two given speeds arithmetically rather than solving through equations. Another is to ignore the starting time and attempt to work only with differences, which can be confusing. Setting up the equations from the arrival times keeps the reasoning clear.

Final Answer:
The girl must cycle at 12 km/h to arrive at 12 noon.