Problem 23 - Elementary and Intermediate Algebra

Question

Tour De France. Maurice Garin of France won the first Tour de France bicycle road race in \(1903 .\) In 2005 , American Lance Armstrong won his seventh consecutive Tour de France. Armstrong's average speed in 2005 was 10 mph faster than Garin's in \(1903 .\) In the time it took Garin to ride 80 miles, Armstrong could have ridden 130 miles. Find each cyclist's average speed.

Step-by-Step Solution

Answer

Garin's speed: 16 mph, Armstrong's speed: 26 mph.

Step 1: Set Up Variables for Speeds

Let Garin's average speed in 1903 be \( v \) mph. Consequently, Armstrong's average speed in 2005 is \( v + 10 \) mph, since it is 10 mph faster than Garin's speed.

Step 2: Express Distances in Terms of Speeds

We are given that Garin covers 80 miles in the same time that Armstrong could cover 130 miles. Let \( t \) be the time it takes for Garin to cover 80 miles, then \( t = \frac{80}{v} \). Similarly, Armstrong would travel \( t = \frac{130}{v+10} \).

Step 3: Set Up Equation

Since the times are equal for both cyclists in their respective distance travels, set the two expressions for time equal to each other: \[ \frac{80}{v} = \frac{130}{v+10} \]

Step 4: Solve for Garin's Speed

Cross-multiply to clear the fractions: \[ 80(v + 10) = 130v \]Simplify and solve for \( v \): \[ 80v + 800 = 130v \]\[ 800 = 50v \]\[ v = 16 \]Garin's average speed is 16 mph.

Step 5: Calculate Armstrong's Speed

Using the expression for Armstrong's speed, which is 10 mph faster than Garin's, his speed is:\[ v + 10 = 16 + 10 = 26 \]Thus, Armstrong's average speed is 26 mph.

Key Concepts

Average Speed Calculation

In problems involving average speed, it's crucial to understand the relationship between distance, speed, and time. The formula to remember is:

• Speed = Distance / Time

By rearranging this formula, we can also express time in terms of distance and speed:

• Time = Distance / Speed

In the context of the Tour de France example, Garin's cycling average speed in 1903 was expressed as a variable, while Armstrong's speed was given as 10 mph faster. By calculating their speeds over different distances but in constant time frames, we were able to derive a meaningful comparison.
The art of calculating average speed involves maintaining clarity on which variable represents speed, distance, or time, to avoid confusion. Once you find the time it takes for one rider to travel a certain distance, the other rider's distance-time relationship can provide the missing piece for complete calculation.

Equation Solving

Solving equations, especially those that arise in word problems, is about transforming a real-world scenario into a solvable mathematical expression. In this exercise, equal travel times formed the basis of the equation:

• \[ \frac{80}{v} = \frac{130}{v+10} \]

This equation matched Garin's distance-to-speed ratio with Armstrong's, considering their known speed difference. The equation equates two expressions, representing the time taken for both cyclists covering their respective distances.
The first step in solving this equation is cross-multiplication. This method helps eliminate fractions, making the equation simpler:

• \[ 80(v + 10) = 130v \]

Once simplified to \[ 80v + 800 = 130v \], basic algebraic manipulation guides you to the solution where calculating for \( v \) yields Garin's speed as 16 mph.

Setting Up Variables

Setting up variables is the crucial first step in tackling algebraic word problems. To convert descriptive problems into equations, identify the unknowns. These unknowns become your variables. In the Tour de France problem, Garin's average speed was unknown, so we represented it with \( v \).
Since Armstrong's speed was given as 10 mph faster than Garin's, it was represented as \( v + 10 \). By setting up variables, you transform narrative statements into mathematical symbols, creating equations that can be solved.

• Define what each variable represents clearly.
• Connect these variables with known conditions or comparisons in the problem.

This structure not only organizes the information but also sets the stage for equation formulation. Correctly setting up variables simplifies later solving steps and increases clarity when checking solutions against the problem scenario.