Problem 54 - Prealgebra

Question

The final standings of a hockey league are shown. A win is worth three points, and a tie is worth 1 point. Zero points are given for a loss. How many points did the Wildcats have? $$\begin{array}{|c|c|c|c|} \hline \text { Team } & \text { Wins } & \text { Losses } & \text { Ties } \\\\\hline \text { Knights } & 14 & 9 & 7 \\\\\hline \text { Huskies } & 11 & 9 & 10 \\\\\hline \text { Wildcats } & 10 & 9 & 11 \\\\\hline \text { Mustangs } & 9 & 10 & 11 \\\\\hline \text { Panthers } & 10 & 14 & 6 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The Wildcats have 41 points.

Step 1: Understanding Points System

Each win earns a team 3 points and each tie results in 1 point. Losses do not earn any points.

Step 2: Calculate Points from Wins

The Wildcats have 10 wins. Each win is worth 3 points. Therefore, the points earned from wins are: \(10 \times 3 = 30\) points.

Step 3: Calculate Points from Ties

The Wildcats have 11 ties. Each tie is worth 1 point. Therefore, the points earned from ties are: \(11 \times 1 = 11\) points.

Step 4: Sum Total Points

Add the points from wins and ties to get the total points for the Wildcats: \(30 + 11 = 41\) points.

Key Concepts

Points System in Sports Leagues

Sports leagues, like hockey, often use a points system to determine team standings. This system helps rank teams based on their performance during the season. A well-known points system awards:

• 3 points for a win
• 1 point for a tie
• 0 points for a loss

The idea is to reward teams for winning games and still recognize them for ties. It encourages competitive play, as winning provides the maximum points possible. A team that accumulates more points ranks higher in the league standings. In our exercise, understanding how points are awarded is crucial for calculating the Wildcats' total score.

Basic Arithmetic Techniques

Basic arithmetic is foundational to everyday problem-solving. In the context of calculating sports standings, it involves simple operations such as multiplication and addition. Let's revisit the Wildcats' performance:
1. Calculate points from wins: The Wildcats had 10 wins. Given each win equals 3 points, we use multiplication: \(10 \times 3 = 30\) points.
2. Calculate points from ties: They also had 11 ties. With each tie worth 1 point, this is a simple multiplication: \(11 \times 1 = 11\) points.
3. Add it all up: Sum the points from wins and ties. Use addition to find the total: \(30 + 11 = 41\) points.
These operations are simple yet powerful for calculating totals in scenarios like sports leagues.

Mastering Problem-Solving Skills

Problem-solving is not just about finding an answer; it's about understanding the steps taken to reach that conclusion. It teaches logical thinking and encourages a structured approach to tackle questions. Here's how the problem-solving process worked in the exercise:

• Identify the problem: The goal was to find the total points for the Wildcats.
• Understand the elements: Recognize the points assigned for wins and ties and that losses do not count.
• Plan your approach: Decide to break down the problem by first calculating points from wins, then points from ties.
• Execute your plan: Use arithmetic to find the points from wins and ties.
• Review your solution: Add the points together and double-check the calculations.

Approaching problems with these steps can simplify complex tasks, making problem-solving a quick and efficient process.