Alan and Brigitte are beginning a bike race. Alan begins riding at 12 miles per hour, and ten minutes later Brigitte begins riding at 15 miles per hour. How many minutes will it take her to catch up to Alan?

This type of puzzle is a classic example of a "catch-up" problem, which is solved by figuring out the relative speed between the two moving objects. Before Brigitte even starts pedaling, Alan has a 10-minute head start. Traveling at 12 miles per hour, which is equivalent to a mile every five minutes, Alan has already created a 2-mile gap. This is the distance Brigitte must close.

To find out how long it will take, we need to look at the difference in their speeds. Brigitte travels at 15 miles per hour, while Alan is moving at 12 miles per hour. This means Brigitte is closing the distance between them at a rate of 3 miles per hour (15 mph - 12 mph). This is their relative speed. Now, the problem is simple: how long does it take to cover that initial 2-mile gap while traveling at 3 miles per hour?

Using the formula Time = Distance / Speed, we can calculate the time it will take for Brigitte to catch up. Dividing the 2-mile gap by her relative speed of 3 mph gives us 2/3 of an hour. Since the question asks for the time in minutes, we multiply 2/3 by 60, which gives us the final answer of 40 minutes. This same principle is used in fields like aviation and marine navigation to determine interception courses and avoid collisions.