Problem 38 - Prealgebra

Question

Use the following information. American Lance Armstrong won the 2005 Tour de France, completing the 2102 -mile race in 83 hours, 36 minutes, 2 seconds. Armstrong also won the 2003 Tour de France. He completed the 2125 -mile race in 80 hours, 2 minutes, 8 seconds. Without calculating, determine which race was completed with a faster average speed. Explain.

Step-by-Step Solution

Answer

The 2003 Tour de France was completed at a faster average speed.

Step 1: Understand the Problem

We are given two races, each with its total distance and total time taken by Lance Armstrong to complete them. We need to determine which race was completed at a faster average speed without performing calculations.

Step 2: Recall Average Speed Formula

Average speed is calculated as total distance divided by total time. Therefore, to determine which race was faster, we compare the ratio of distance to time for each race.

Step 3: Compare Time and Distance Proportion

For the 2005 race, Lance Armstrong traveled 2102 miles in a longer time (83 hours, 36 minutes, 2 seconds) compared to the 2003 race distance of 2125 miles but with a shorter time (80 hours, 2 minutes, 8 seconds). Although the 2003 race is longer, the fact that he took less time generally indicates a higher average speed for the 2003 race.

Step 4: Conclusion

Since Lance Armstrong took a shorter amount of time to cover a slightly larger distance in 2003, the average speed for the 2003 Tour de France would be faster than that of the 2005 race.

Key Concepts

Average Speed

Average speed is a concept that helps us understand how fast something is moving overall over a period. We calculate average speed by dividing the total distance traveled by the total time taken. If you know the distance and time, you can easily find out the average speed without needing a calculator.

• Formula: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)
• This formula gives you the speed as distance per unit of time, like miles per hour (mph).

Understanding this formula is crucial because it simplifies complex journeys into a single, understandable number. In our example of the Tour de France races, though the exact calculations weren't done, recognizing that the distance to time ratio impacts speed helps in drawing conclusions.

Distance and Time Relationships

Distance and time are closely linked in understanding speed. When we look at a journey, the distance is how far you've traveled, and time is how long it took. Together, they help determine speed. In the context of the Tour de France example, comparing these two aspects gives insight into the performance.
If you travel a greater distance in less time, your speed is likely higher. Conversely, if you take more time to cover a shorter distance, your speed is lower. This was key in analyzing Lance Armstrong's races. Despite the 2003 race being longer, the shorter time suggested a faster speed.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to solve problems. It is the backbone of making informed decisions in mathematics. When solving problems like the one in our exercise, mathematical reasoning helps analyze information even without exact calculations.
Here, using reasoning, we deduced that the 2003 race was completed at a higher speed based on an understanding of distance-to-time relationships. This reasoning allowed us to make a confident conclusion using the information given, showcasing how powerful logical thinking can be.

Comparison of Ratios

Ratios are a way to compare two quantities mathematically. In speed-related problems, comparing ratios helps us determine which of two or more scenarios is faster or slower. Think of ratios as a comparison of how much distance is covered for each unit of time.
By comparing the ratios from the two Tour de France races, we identified which ratio of distance to time was larger. This implies higher speed without needing exact arithmetic. Ratios simplify comparisons in various cases and are valuable in assessing performance, such as determining that a shorter time for a longer distance typically means greater speed.