Problem 39 - Prealgebra

Question

Use the table and the following information. A national poll ranks college football teams using votes from sports reporters. Each vote is worth a certain number of points. Suppose that Penn State University receives 50 first-place votes, 7 second-place votes, 4 fourth-place votes, and 3 tenth-place votes. $$\begin{array}{|c|c|}\hline\ \text { Number of Points for Each Vote } \\\\\hline \text { Vote } & \text { Points } \\\\\hline \text { 1st place } & 25 \\\\\hline \text { 2nd place } & 24 \\\\\hline \text { 3rd place } & 23 \\\\\hline \text { 4th place } & 22 \\\\\hline \text { 5th place } & 21 \\\\\hline \vdots & \vdots \\\\\hline \text { 25th place } & 1 \\\\\hline\end{array}$$ Find the total number of points.

Step-by-Step Solution

Answer

1554 points.

Step 1: Understand the information

We are given that Penn State University received a certain number of votes for different ranks. Each rank has an associated number of points. We need to calculate the total points based on these ranks.

Step 2: Calculate points for first-place votes

Number of first-place votes = 50. Points for each first-place vote = 25. Calculate the points: \[ 50 \times 25 = 1250 \] points.

Step 3: Calculate points for second-place votes

Number of second-place votes = 7. Points for each second-place vote = 24. Calculate the points: \[ 7 \times 24 = 168 \] points.

Step 4: Calculate points for fourth-place votes

Number of fourth-place votes = 4. Points for each fourth-place vote = 22. Calculate the points: \[ 4 \times 22 = 88 \] points.

Step 5: Calculate points for tenth-place votes

Number of tenth-place votes = 3. Points for each tenth-place vote = 16. Calculate the points: \[ 3 \times 16 = 48 \] points.

Step 6: Sum all the points

Add all the points calculated from different votes to find the total number of points:\[ 1250 + 168 + 88 + 48 = 1554 \].

Key Concepts

Rank-Based Points System

In the rank-based points system used for ranking college football teams, each vote received is assigned a specific number of points based on the rank given. This system ensures that higher-ranked votes contribute more significantly to a team's overall score than lower-ranked votes. For example, a first-place vote is more valuable than a tenth-place vote.

The points are set up hierarchically, as shown in the exercise:

• 1st place - 25 points
• 2nd place - 24 points
• 3rd place - 23 points
• ...
• 25th place - 1 point

This system is similar to how ranks are often calculated in contests and competitions, where the aim is to reward those who achieve higher ranks with more points, thus promoting competitiveness.

Mathematical Calculations

Mathematical calculations are at the core of determining the total points in this ranking system. Each rank's votes need to be multiplied by their respective points to find out how much each rank contributes to the overall score.

In our example, the calculations for each rank were as follows:

• First-place: 50 votes 25 points = 1250 points
• Second-place: 7 votes 24 points = 168 points
• Fourth-place: 4 votes 22 points = 88 points
• Tenth-place: 3 votes 16 points = 48 points

After each individual calculation, we sum these products to find the total points accumulated by the team. Such arithmetic operations are fundamental in analyzing data and could also be applied in various contexts, such as calculating grades or averaging expenses.

Arithmetic Operations

To solve the problem, basic arithmetic operations are employed, which include multiplication and addition, integral parts of prealgebra. The processes involve multiplying the number of votes by the points assigned to each rank and then summing these products to find the total.

Here's a breakdown of the operations used:

• Multiplication: Used to compute the points for each rank category by multiplying the votes with corresponding point values.
• Addition: All results from multiplication are summed up to obtain the final total points.

Understanding these operations is essential, as they form the basis for more complex mathematical concepts encountered in further algebra studies. Prealgebra focuses on grounding students in these fundamental arithmetic operations, making exercises like these an excellent practice for reinforcing math fluency.