Problem 38 - 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. $$\begin{array}{|c|c|}\hline\ \text { Number of Points for Each Vote } \\\\\hline \text { Vote } & \text { Points } \\\\\hline \text { 1st place } & 25 \\\\\hline \text { 2nd place } & 24 \\\\\hline \text { 3rd place } & 23 \\\\\hline \text { 4th place } & 22 \\\\\hline \text { 5th place } & 21 \\\\\hline \vdots & \vdots \\\\\hline \text { 25th place } & 1 \\\\\hline\end{array}$$ Write an expression for the number of points that Penn State receives.

Step-by-Step Solution

Answer

Penn State receives 1554 points.

Step 1: Identify the Points for Each Vote Type

According to the given table, each type of vote is worth a different number of points. We have the following values: - 1st place vote is worth 25 points - 2nd place vote is worth 24 points - 4th place vote is worth 22 points - 10th place vote is worth 16 points.

Step 2: Calculate Points for 1st Place Votes

Penn State received 50 first-place votes, and each of these votes is worth 25 points. Thus, the total points from first-place votes are calculated as:\[50 \times 25 = 1250\]

Step 3: Calculate Points for 2nd Place Votes

Penn State received 7 second-place votes, with each vote worth 24 points. The total points from these are:\[7 \times 24 = 168\]

Step 4: Calculate Points for 4th Place Votes

Penn State received 4 fourth-place votes, with each vote worth 22 points. Thus, the total points from fourth-place votes are:\[4 \times 22 = 88\]

Step 5: Calculate Points for 10th Place Votes

For 10th place votes, Penn State received 3 votes, each worth 16 points. The total points from these votes are:\[3 \times 16 = 48\]

Step 6: Sum All Points to Get Total

Now, sum up all the points from each type of vote to find the total number of points Penn State receives:\[1250 + 168 + 88 + 48 = 1554\]

Step 7: Write the Expression

Based on the steps above, the expression for the total points received by Penn State is:\[50 \times 25 + 7 \times 24 + 4 \times 22 + 3 \times 16\] This corresponds to a total points of 1554.

Key Concepts

Voting System

In many competitive scenarios, especially in sports, different voting systems are set up by organizations to rank teams or players. A voting system determines how votes are collected and calculated to produce a final outcome.
For college football rankings, sports reporters often cast votes for teams, with each vote assigning a particular number of points based on the team's rank in their ballot. This type of system is a weighted voting system, where higher-ranking positions yield more points than lower-ranking ones.
This ensures that while a team’s consistent performance at the top is valued, varied performances across ranking positions still contribute to their total score. Understanding how voting systems distribute points is key to figuring out how rankings are determined and updated as the season progresses.

Points Calculation

When working with a points system, it’s essential to calculate the total points received accurately. Each vote counts towards the team's total score, depending on the rank:

• Each first-place vote adds 25 points.
• Each second-place vote adds 24 points.
• Continuing down, such as fourth-place adding 22 points, and so forth.

To calculate the number of points a team gets from each type of vote, multiply the number of votes by the points per vote.
For example, if a team gets 50 first-place votes, you multiply 50 by 25, resulting in a total of 1250 points from those votes alone.
Manual calculations ensure accuracy and allow you to clearly see how each vote contributes to the final score, which is essential for transparency and to build strategies around accumulating points.

Multiplication and Addition

To determine a team's total score in a voting system using points, you need to master some basic math operations, such as multiplication and addition. Each category of vote requires you to calculate a product of votes and their assigned point value:

• Multiply the number of votes by the points per vote category (e.g., 50 votes for first place, each worth 25 points, equals 50 times 25, which is 1250 points).
• Do the same for each category of votes that a team receives.

After calculating the partial scores from each category of votes, sum these products:
Add up all of the multiplication results to find the total points (e.g., add all calculated results from first, second, fourth, and tenth-place votes).
This blend of multiplication and addition ensures that you compute the total points comprehensively, leading to an accurate score for the team's rank.

Ranking Method

Once the total points for each team are calculated, these scores are used to establish a ranking or order. A ranking method involves arranging teams in a list from highest to lowest based on their total points.
The team with the highest points is ranked first, followed by teams with lower points in descending order. The method gives a clear and concise way to see which team stands out among peers and is especially helpful in competitive environments.
The importance of a ranking method is not only to highlight the best-performing teams but also to motivate other teams to improve and aim for higher rankings. Rankings are dynamic; they can change with each new set of votes, encouraging consistent performance.
Understanding the mechanism of ranking helps learners grasp how relative positions in a list can change over time based on performance and voting outcomes.