Problem 104 - Elementary and Intermediate Algebra

Question

Politics. Six months before an election, the incumbent trailed the challenger by 18 points. To overtake her opponent, the incumbent decided to use a four- part strategy. Each part of the plan is shown below, with the expected point gain. With these gains, will the incumbent overtake the challenger on election day? \(\cdot\) TV ads \(\quad+10\) pts \(\quad \cdot\) Union endorsement \(\quad+2\) pts \(\cdot\) Voter mailing \(+3\) pts \(\quad \cdot\) Telephone calls \(\quad+1\) pts

Step-by-Step Solution

Answer

The incumbent remains 2 points behind and does not overtake the challenger.

Step 1: Initial Point Deficit

The incumbent is initially trailing the challenger by 18 points. This is the starting point deficit that the incumbent needs to overcome.

Step 2: Compute Total Expected Gain

Add up all the point gains expected from the four-part strategy: TV ads (+10 points), Union endorsement (+2 points), Voter mailing (+3 points), and Telephone calls (+1 point).\[ 10 + 2 + 3 + 1 = 16 \] The total expected gain is 16 points.

Step 3: Calculate Net Points

Subtract the total expected gain from the initial point deficit to determine if the incumbent overtakes the challenger.\[ 18 - 16 = 2 \] After applying the strategy, the incumbent is still 2 points behind.

Step 4: Conclusion

Since the incumbent still trails by 2 points despite the strategy, the incumbent does not overtake the challenger on election day according to these calculations.

Key Concepts

Step-by-Step Solution

A step-by-step solution is a powerful approach to problem-solving in algebra. It’s like having a roadmap to navigate through the problem. By breaking down the problem into manageable steps, you can solve even the most complex problems with ease. Here's a quick rundown of why this method is effective:

• Clear Structure: Each step is like a chapter in a book, making the overall story easier to understand.
• Enhanced Focus: Focusing on one step at a time helps prevent feeling overwhelmed by the entire problem.
• Error Identification: Mistakes can often be easily spotted and corrected in the earlier steps.

In our example, the solution begins by recognizing the initial point deficit as a critical foundation. Each subsequent step systematically addresses and abstracts complex calculations, ensuring clarity throughout the solution process.

Addition and Subtraction of Integers

In algebra, addition and subtraction of integers are fundamental operations. These operations are not only vital for solving algebraic equations but also for everyday calculations. Here's a simple breakdown:

• Addition: When adding integers, you move right on the number line. For example, starting from a deficit of -18 and adding 16, you move towards zero, reaching -2.
• Subtraction: Subtraction is like adding a negative number. It's about identifying how much you need to add to regain balance. In the exercise, you subtract the gain from the initial deficit to assess the leftover gap.

Understanding these operations is pivotal for calculating deficits and surpluses, especially in situations like political campaigns where every point counts.

Political Campaign Strategy

A political campaign strategy is crucial for any candidate aiming to win an election. It involves a series of planned actions designed to influence voter opinions and secure victory. In this problem, the campaign strategy is broken down into four specific actions:

• TV Ads: Widely reach potential voters, raising awareness and potentially gaining 10 points.
• Union Endorsement: Gaining union support which could add 2 points, a direct influence by endorsement.
• Voter Mailing: Personalized communication that may gain 3 points by directly reaching people’s homes.
• Telephone Calls: Personal outreach for immediate engagement, driving up support by 1 point.

By leveraging various outreach methods, each with an expected impact, the candidate attempts to close the gap with the opponent effectively.

Mathematical Reasoning

Mathematical reasoning is about using logical thinking to evaluate a problem and derive a conclusion. It combines numeric operations with strategic thinking, essential in determining the effectiveness of actions within a political context. Here's why it's important:

• Evaluation: Assessing initial conditions and final outcomes to see if your strategy works.
• Logical Steps: Developing hypotheses (like expecting a point gain) and testing them through calculations.
• Interpreting Results: Understanding the outcome of calculations and making informed decisions.

In our example, mathematical reasoning allows the candidate to determine that despite effort and gains, they are still behind, leading to informed decisions on future strategy adjustments.