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Problem solving in physics revolves around the fundamental process of observing a phenomenon, translating it into a mathematical model, applying relevant equations or principles, and then solving for the desired unknowns. The effectiveness of this approach hinges on a systematic method and a clear understanding of the physical concepts involved. In the case of the bicyclists with varying speeds, the problem prompts us to determine aggregate measures of their motion, namely the average and rms speed.

Breaking down complex problems into manageable steps is a key strategy. For example, compiling and analyzing the given data, considering the proper formulas, and performing calculations attentively are imperative. Always keep track of units since they provide crucial insight into the physical quantities being measured. Moreover, rechecking each step can prevent errors and reinforce understanding, an advice especially useful in multi-step exercises like the one demonstrated with the bicyclists' speeds.

Average speed is a fundamental concept in mechanics that describes how fast an object moves over a certain distance without considering the intricacies of varying speeds during the trip. It is essentially the total distance traveled divided by the total time taken. In scenarios where objects move at different speeds for equal time intervals, like the bicyclists in our problem, the average speed can be found by summing all individual speeds and dividing by the number of intervals or entities involved.

To illustrate, if each bicyclist represents a time interval moving at a distinct speed, calculating the average speed involves adding up all the individual speeds and then dividing by the total number of bicyclists. As shown in the solution, this straightforward method yields the overall average speed, providing a simple yet informative measure of their collective pacing.

The root mean square (rms) speed is a statistical measure used in physics to describe the magnitude of a varying quantity. It gives a meaningful average when the data involves both magnitude and direction, common in fields like electromagnetism and thermodynamics. For speeds, the rms speed has a specific significance in the context of particles in a gas where speeds can vary in magnitude and direction. However, it can also apply to cases like our bicyclists, providing a different perspective on average speed.

To calculate rms speed, each individual speed value is squared, which removes the direction aspect for vectors and amplifies larger values, then these squares are averaged, and finally, the square root of this average is taken to restore the quantity to its original units. The exercise demonstrates this process, highlighting the subtle but important differences between average and rms speed — while both are types of averages, rms speed tends to give higher values when the range of speeds is large, a detail often overlooked in preliminary physics education.