Solving a Mathematical Puzzle: John, Kevin, and Tod's Points

In a recent competition, the scores of participants John, Kevin, and Tod were a combination of algebraic relations and numerical values. This problem not only requires an understanding of basic algebra but also the application of systems of equations to determine the specific scores of each individual. Let's delve into the step-by-step solution to find out how many points John, Kevin, and Tod each scored.

Step-by-Step Solution

The problem states that a total of 83 points were scored by John, Kevin, and Tod during a competition. Additionally, it is given that:

Kevin scored 13 points less than John. Tod scored 3 times as many points as John.

We can represent these statements algebraically. Let's denote:

John's score as x Kevin's score as 3x - 13 Tod's score as 3x

According to the problem, the sum of their scores is 83:

x (3x - 13) 3x 83

Step 1: Combining Like Terms

First, let's combine the like terms on the left-hand side of the equation.

x 3x 3x - 13 83

7x - 13 83

Step 2: Isolating the Variable

To solve for x, we need to isolate the variable on one side of the equation. We start by adding 13 to both sides to eliminate the -13 term.

7x - 13 13 83 13

7x 96

Next, divide both sides by 7 to solve for x.

x 96 / 7

x 10

Step 3: Finding Each Individual's Score

Now that we know John's score is 10, we can find Kevin's and Tod's scores using the initial relationships we established.

John's score, x 10 Kevin's score, 3x - 13 3(10) - 13 30 - 13 17 Tod's score, 3x 3(10) 30

Therefore, John scored 10 points, Kevin scored 43 points, and Tod scored 30 points.

Verification

Let's verify the solution to ensure all conditions of the problem are met:

John (x) Kevin (3x - 13) Tod (3x) 83

10 43 30 83

The equation holds true, confirming that our solution is correct.

Conclusion

In summary, we used algebra to solve a real-world problem involving competition scores. By setting up and solving the equation, we determined that John scored 10 points, Kevin scored 43 points, and Tod scored 30 points. Understanding and applying mathematical concepts like systems of equations can help solve complex real-world scenarios.

Key Takeaways

Understanding the problem statement and converting it into algebraic equations is crucial before solving. Combining like terms, isolating the variable, and substituting the value back into the equation are essential steps in solving the problem. Verification is vital to ensure the solution is correct and meets all the conditions of the problem.

Related Questions

What are Mathematical Equations? What is Algebraic Problem Solving? How to Calculate Competition Scoring?