Problem 105 - Elementary and Intermediate Algebra

Question

NASCAR. Complete the table below to determine how many points the third and fourth place finishers were behind the leader. $$ \begin{array}{|c|c|c|c|} \hline & {2009 \text { Final Driver Standings }} & {} & {} \\ \hline & {} & {\text { Points }} & {\text { Points behind }} \\ \hline \text { Rank } & {\text { Driver }} & {} & {\text { leader }} \\ \hline 1 & {\text { Jimmie Johnson }} & {6,652} & {. .} \\ \hline 2 & {\text { Mark Martin }} & {6,511} & {-141} \\ \hline 3 & {\text { Jeff Gordon }} & {6,473} & {} \\ \hline 4 & {\text { Kurt Busch }} & {6,446} & {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Jeff Gordon is 179 points behind, and Kurt Busch is 206 points behind the leader.

Step 1: Understanding the Table

The table shows the final NASCAR driver standings for 2009. It includes the rank, driver's name, their points, and how many points they are behind the leader.

Step 2: Identify Leader's Points

From the table, we see that Jimmie Johnson is the leader with 6,652 points. This will be our reference point to calculate how many points other drivers are behind.

Step 3: Calculate Points Behind for Jeff Gordon

To find out how many points Jeff Gordon is behind Jimmie Johnson, subtract Jeff Gordon's points from Jimmie Johnson's points: \[ ext{Points behind for Jeff Gordon} = 6,652 - 6,473 = 179 \]

Step 4: Calculate Points Behind for Kurt Busch

Similarly, calculate how many points Kurt Busch is behind Jimmie Johnson by subtracting Kurt Busch's points from Jimmie Johnson's points: \[ ext{Points behind for Kurt Busch} = 6,652 - 6,446 = 206 \]

Step 5: Complete the Table

Fill in the 'Points behind leader' for Jeff Gordon and Kurt Busch with the calculated values: - Jeff Gordon is 179 points behind the leader. - Kurt Busch is 206 points behind the leader.

Key Concepts

Subtraction in Context

Understanding subtraction in real-world contexts can be very useful. To solve the problem given in the NASCAR table, we need to find how far behind certain drivers are from the leader using subtraction. By identifying the leader's total points, we can subtract this from each driver's point total to find out how many points they are behind:

- For Jeff Gordon, we subtract his total of 6,473 points from the leader's 6,652 points. This calculation results in him being 179 points behind.
- Kurt Busch's situation is similar, with his points (6,446) being subtracted from the leader's total (6,652), resulting in a 206-point lag.

This method helps us understand the standing of each driver in relation to the leader, providing a clear picture of the points gap, simply using subtraction.

Mathematical Tables

Mathematical tables are a powerful way to organize and interpret data. They allow us to visually display numbers and relationships, making comparisons much easier. In this exercise, the table contains race driver names, their ranks, points, and points behind the leader. Steps to effectively interpret a mathematical table include:

• Identifying the headings, which specify what each column represents in terms of data.
• Locating the relevant figures to execute calculations,—here, both points and the leader's reference point are essential.
• Using the organized data to perform step-by-step calculations, finding numeric relationships such as point differences.

This organization aids in drawing conclusions and performing calculations necessary for problem-solving.

Data Interpretation

Data interpretation involves making sense of numerical data, often in the form of tables and charts, to extract meaningful insights. In NASCAR standings, data like ranks, driver names, and points help us see how drivers perform compared to each other. Key steps in data interpretation include:

• Examining data to decide what questions need to be answered (e.g., how far behind each driver is).
• Applying mathematical operations to find answers (subtraction in this case).
• Understanding the practicality of results—knowing one driver is 179 points behind while another is 206 points behind can inform strategy or underline performance differences.

By looking at just numbers and data presented in an organized way, we can draw impactful conclusions about the standings and implications in NASCAR rankings.

Real-world Applications of Algebra

Algebra isn’t just theoretical; it’s a cornerstone of solving real-world problems. Let's see how this exercise fits in the wider context of everyday applications:

- **Point Comparisons:** In motorsports like NASCAR, understanding rankings and margins separates merely completing a race from strategic positioning throughout a season.
- **Budgeting and Forecasting:** Companies and individuals use similar techniques to keep track of financial standings. Using subtraction and other basic algebraic operations, we can clearly map expenses against income or other predefined figures.
- **Sports Analytics:** Beyond NASCAR, similar algebra principles help in analyzing team or individual sports performances to better prepare tactics.

In all these, algebraic problem-solving often helps in making data-driven decisions rather than just guesses.