Robert travels by cycle towards a point A. He will reach A at 2:00 p.m. if he cycles at 10 km/h, and he will reach A at 12:00 noon if he cycles at 15 km/h. At what constant speed must he cycle in order to reach A exactly at 1:00 p.m.?

Introduction / Context:
This question is a classic example of using two scenarios with different speeds and arrival times to deduce the total distance and starting time. Once the distance is known, we can find the speed needed to arrive at a new specified time. Problems of this type are common in aptitude tests and help reinforce the relation between distance, speed, and time in real life scheduling situations.

Given Data / Assumptions:

• If Robert cycles at 10 km/h, he reaches A at 2:00 p.m.
• If he cycles at 15 km/h, he reaches A at 12:00 noon.
• The starting time from his home is the same in both cases.
• The distance from his home to point A is fixed and does not change.
• We assume straight line travel and constant speeds.

Concept / Approach:
Let D be the distance from Robert's home to A, and let t10 and t15 be the travel times at 10 km/h and 15 km/h respectively. We can express D as 10 * t10 and also as 15 * t15. The difference between t10 and t15 is 2 hours because 2:00 p.m. is 2 hours after 12:00 noon. Solving this system we get both D and the starting time. Once the distance is known, we compute the time available to arrive at 1:00 p.m. and then find the required speed using speed = distance / time.

Step-by-Step Solution:
Let t10 be the travel time at 10 km/h. Let t15 be the travel time at 15 km/h. Distance D = 10 * t10 = 15 * t15. Arrival at 2:00 p.m. and 12:00 noon from the same start implies t10 - t15 = 2 hours. From D = 10 * t10 and D = 15 * t15, we have 10 * t10 = 15 * t15. Thus, t10 = (3 / 2) * t15. Substitute into t10 - t15 = 2: (3 / 2) * t15 - t15 = 2. This simplifies to (1 / 2) * t15 = 2, so t15 = 4 hours. Then t10 = t15 + 2 = 6 hours. Distance D = 10 * t10 = 10 * 6 = 60 km. If Robert wants to arrive at 1:00 p.m., the travel time from his start must be 5 hours (since he starts 6 hours before 2:00 p.m., that is at 8:00 a.m.). Required speed = distance / time = 60 / 5 = 12 km/h.

Verification / Alternative check:
We can verify the starting time. At 10 km/h, Robert takes 6 hours and reaches at 2:00 p.m., so he must have started at 8:00 a.m. At 15 km/h, he takes 4 hours and reaches at 12:00 noon, also starting at 8:00 a.m. This is self consistent. To arrive at 1:00 p.m. starting from 8:00 a.m., he has 5 hours of travel time. Traveling 60 km in 5 hours clearly requires a speed of 12 km/h, which confirms the answer.

Why Other Options Are Wrong:
Speeds like 20 km/h, 18 km/h, 16 km/h, or 14 km/h would not allow Robert to both cover the 60 km distance and arrive exactly at 1:00 p.m. For example, at 20 km/h, he would arrive much earlier, and at 16 km/h he would not arrive on time. Only 12 km/h exactly matches the distance and required travel time.

Common Pitfalls:
Some students incorrectly average the two given speeds or the two arrival times instead of solving the equations properly. Others forget that the difference between arrival times reflects the difference between the two travel durations. Clearly writing the equations for distance and time in each scenario avoids these mistakes and leads to the correct required speed.

Final Answer:
Robert must travel at a constant speed of 12 km/h to reach point A at 1:00 p.m.