Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Investment Portfolio A total of \(\$ 25,000\) is invested in two corporate bonds that pay \(9.5 \%\) and \(14 \%\) simple interest. The total annual interest is \(\$ 3050 .\) How much is invested in each bond?

Step-by-Step Solution

Answer

The amount which should be invested in the bond with the interest rate of \(9.5\%\) is \(x\) dollars and the bond with the interest rate of \(14\%\) is \(y\) dollars. The values will be calculated by solving the equations.

Step 1: Formulate the equations

Let's let \(x\) be the amount invested at \(9.5\%\) and \(y\) be the amount invested at \(14\%\). The problem provides two pieces of information which can be translated into two equations: 1. The total money invested is \$25000, which gives the equation \(x + y = 25000\). 2. The total interest from both investments is \$3050 , which gives the equation \(0.095x + 0.14y = 3050\).

Step 2: Solve the system of equations

Setting up a system of linear equations allows us to solve for both \(x\) and \(y\). Solve the first equation for \(x\) to find \(x = 25000 - y\). Substitute this into the second equation to get \(0.095(25000 - y) + 0.14y = 3050\). Simplifying this will give a value for \(y\) after calculating.

Step 3: Final calculation

Substitute \(y\) back into the initial equation, \(x + y = 25000\), to find the value of \(x\).

Key Concepts

System of Linear Equations

Understanding how to construct and solve a system of linear equations is crucial when tackling problems involving two or more variables that are related in some way. In this exercise, we are given a scenario with a total investment of \$25,000 across two bonds with different interest rates. By establishing variables, \(x\) and \(y\), for the amounts invested in each bond and creating two separate equations that describe the relationships of these amounts, we set up our system of linear equations.

Our first equation, \(x + y = 25000\), comes from the total investment amount. The second equation, \(0.095x + 0.14y = 3050\), represents the total annual interest earned from both bonds. These are linear because each variable is raised to the power of 1, and they are graphically represented by straight lines in coordinate space. The intersection point of these lines gives us the solution to the system - the exact amounts invested in each bond. Using the method of substitution or elimination, we can find these values, providing us with a practical application of algebra in financial planning.

Investment Analysis in Algebra

Algebraic techniques are incredibly useful for analyzing investments and understanding how different financial variables interact. This exercise demonstrates an essential application: determining how much money to allocate in different investments based on the return rates.

In the context of simple interest, the interest earned is a direct result of the rate applied to the principal amount. The algebraic model we've built with our linear equations is a simplification of the real-world process of investment analysis. It enables us to split a total investment into parts that yield different interest rates, effectively optimizing the returns. By strategically solving these equations, we find the ideal amounts to invest to achieve the desired total interest, optimizing an investment portfolio using algebraic thinking.

Algebraic Problem Solving

Algebraic problem solving involves identifying the relevant information given in a problem, forming algebraic expressions or equations, and then manipulating these equations to find a solution.

In our example, we used the given information about the total amount invested and the interest rates to create equations that accurately represent the situation. The problem-solving process is systematic: define variables, set up equations, use mathematical operations to isolate variables, and finally obtain the solution by substituting the discovered values back into the original equations.

Common Steps in Algebraic Problem Solving

• Understand the problem and identify what is known and what is unknown.
• Choose appropriate variables to represent the unknowns.
• Translate the problem into one or more algebraic equations.
• Solve the equations using algebraic techniques like substitution or elimination.
• Verify the solution by plugging it back into the original context.

By following these steps, algebra allows us to take real-world scenarios and solve them in a structured and logical way, enhancing analytic capabilities across various fields, including finance.