If a bicycle drives 2 miles in 5 minutes, then drives the next 2 miles in 10 minutes, what is the bike's average speed for this entire trip, in miles per hour?

It might seem intuitive to simply average the speeds of the two legs of the journey, but that leads to the wrong conclusion. The first segment (2 miles in 5 minutes) is a speed of 24 miles per hour, while the second segment (2 miles in 10 minutes) is 12 miles per hour. The average of 24 and 12 is 18, which is incorrect. The key to this kind of problem is that average speed is always calculated as the total distance traveled divided by the total time it took.

For this trip, the total distance is straightforward: 2 miles plus 2 miles equals 4 miles. The total time is also simple to add up: 5 minutes plus 10 minutes is 15 minutes. The final step is to make sure the units match what the question asks for, which is miles per hour. Since 15 minutes is a quarter (or 1/4) of an hour, we can now complete the calculation.

Dividing the total distance of 4 miles by the total time of 1/4 hour gives the correct average speed. The reason this method works, and averaging the two speeds doesn't, is that the cyclist spent twice as much time traveling at the slower speed, giving it more "weight" in the final calculation. The total distance over total time formula automatically accounts for this, providing the true average speed for the entire duration of the trip.