The Largest Combination of Policies

Oct 13, 2025 by windailys.com 135 views

Introduction

In this article, we will explore a mathematical problem involving life insurance policies. Peter and Marcia, both aged 34, have a limited budget for life insurance, with each able to pay $650 per year. We aim to find the face value of the largest combination of policies they can buy. This problem requires a deep understanding of mathematical concepts, particularly those related to optimization and linear programming.

Understanding the Problem

Let's break down the problem and understand what's being asked. Peter and Marcia each have a budget of $650 per year for life insurance. They want to buy the largest combination of policies possible, but they need to find the face value of these policies. In other words, they want to maximize the total face value of the policies they can buy within their budget.

Mathematical Formulation

To approach this problem mathematically, we can use the concept of linear programming. We can represent the problem as a linear equation, where the objective function is the total face value of the policies, and the constraints are the budgets of Peter and Marcia.

Let's denote the face value of the policies as x and y for Peter and Marcia, respectively. The objective function can be represented as:

Maximize: x + y

Subject to:

650x ≤ 650 (Peter's budget) 650y ≤ 650 (Marcia's budget) x ≥ 0 (non-negativity constraint for Peter) y ≥ 0 (non-negativity constraint for Marcia)

Solving the Problem

To solve this problem, we can use the graphical method or the simplex method. However, since this is a simple problem, we can solve it by inspection.

From the constraints, we can see that the maximum value of x is 1 (since 650x ≤ 650 implies x ≤ 1), and the maximum value of y is also 1 (since 650y ≤ 650 implies y ≤ 1). However, since they want to buy the largest combination of policies possible, they can buy policies with face values of 1 for both Peter and Marcia.

The Largest Combination of Policies

The largest combination of policies Peter and Marcia can buy is two policies, each with a face value of 1. This means the total face value of the policies they can buy is 2.

Conclusion

In this article, we explored a mathematical problem involving life insurance policies. We used the concept of linear programming to represent the problem and solve it. The largest combination of policies Peter and Marcia can buy is two policies, each with a face value of 1, resulting in a total face value of 2.

References

[1] Linear Programming: A Modern Introduction, by Bradley, Hax, and Magnanti
[2] Optimization Techniques, by Himmelblau

Additional Resources

[1] Khan Academy: Linear Programming
[2] MIT OpenCourseWare: Linear Programming

Mathematical Formulas

Linear Programming: Maximize or Minimize a linear function subject to linear constraints.
Objective Function: The function to be maximized or minimized.
Constraints: The linear inequalities that the variables must satisfy.
Non-Negativity Constraint: A constraint that requires the variable to be non-negative.

Mathematical Concepts

Linear Programming: A method for solving optimization problems with linear objective functions and constraints.
Optimization: The process of finding the best solution among a set of possible solutions.
Linear Function: A function of the form f(x) = ax + b, where a and b are constants.
Linear Inequality: An inequality of the form ax + b ≤ c, where a, b, and c are constants.
Q&A: The Largest Combination of Life Insurance Policies ===========================================================

Introduction

In our previous article, we explored a mathematical problem involving life insurance policies. We used the concept of linear programming to represent the problem and solve it. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the main objective of the problem?

A: The main objective of the problem is to find the face value of the largest combination of policies that Peter and Marcia can buy within their budget.

Q: What are the constraints of the problem?

A: The constraints of the problem are the budgets of Peter and Marcia, which are $650 per year each. Additionally, the face value of the policies must be non-negative.

Q: How did you solve the problem?

A: We used the concept of linear programming to represent the problem and solve it. We set up the objective function as the total face value of the policies and the constraints as the budgets of Peter and Marcia.

Q: What is the largest combination of policies that Peter and Marcia can buy?

A: The largest combination of policies that Peter and Marcia can buy is two policies, each with a face value of 1. This means the total face value of the policies they can buy is 2.

Q: Can Peter and Marcia buy policies with face values greater than 1?

A: No, they cannot buy policies with face values greater than 1. The constraints of the problem do not allow for it.

Q: What if Peter and Marcia want to buy policies with face values less than 1?

A: If Peter and Marcia want to buy policies with face values less than 1, they can do so. However, the total face value of the policies they can buy will be less than 2.

Q: Can you provide an example of how to apply the solution to a real-world scenario?

A: Suppose Peter and Marcia want to buy life insurance policies to cover their funeral expenses. They each have a budget of $650 per year for life insurance. Using the solution we provided, they can buy two policies, each with a face value of 1, which will cover their funeral expenses.

Q: What are some common mistakes to avoid when solving this type of problem?

A: Some common mistakes to avoid when solving this type of problem include:

Not clearly defining the objective function and constraints
Not using the correct mathematical techniques to solve the problem
Not considering all possible solutions
Not checking the validity of the solution