The average speed for the bicyclist for the first 45 miles...

The average speed for the bicyclist for the first 45 miles was 15 miles per hour (mph) and his average speed for the next 50 miles was $25 \mathrm{mph}$. What should his average speed be for the next 85 miles if his overall average speed is to be 18 mph?

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Key Concepts

Time-Distance Relationship

The time-distance relationship is expressed by the equation time = distance/speed. This fundamental concept links speed directly to the distance covered and the time taken. It is essential for setting up equations in problems that require finding unknown speeds or times in order to achieve a predetermined overall average speed.

Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. It is not simply the arithmetic mean of separate speeds when travel involves different segments, each with its own speed and distance. Understanding this concept is crucial for problems where different parts of a journey occur at varying speeds.

Weighted Average

A weighted average takes into account the relative importance of each component in determining the overall average. In the context of speed, when distances or times differ for segments of a trip, the average speed must be computed by weighting each segment by its corresponding travel time or distance, rather than using a simple arithmetic mean of the speeds.

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