Problem 75 - College Algebra and Calculus: An Applied Approach

Question

A total of $$\$ 35,000$$ is invested in two funds paying \(8.5 \%\) and \(12 \%\) simple interest. The total annual interest is $$\$ 3675$$. How much is invested at each rate?

Step-by-Step Solution

Answer

So, \$15000 is invested at 8.5% and \$20000 at 12%

Step 1: Set up the equations

Let \( x\) be the amount invested at \(8.5\%\) and \( y\) the amount at \(12\%\). The total amount invested is given, so we can write the equation \( x+ y = 35000\). The total annual interest is also given so we can write another equation based on the interests which is \(0.085x + 0.12y = 3675\) which represents the total interest earned from both investments.

Step 2: Solve the system of equations

The goal is to solve this system of equations for \( x \) and \( y \). This can be done by substitution or elimination. For this example, the easiest way to do this would be to use substitution. First, rearrange the initial investment equation to solve for \( y \). This gives \( y = 35000 - x\). Then substitute \( y \) in the interest equation. This gives \( 0.085x + 0.12(35000 - x) = 3675\). Simplifying the above equation gives \( x = 15000\). Substituting \( x \) into the initial investment equation gives \( y = 35000 - 15000 = 20000\).

Step 3: Check your answers

To check these answers, substitute \( x = 15000 \) and \( y = 20000 \) into the initial equations. The sums should be equal to the given total investment 35000 and total interest 3675 respectively.

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations with the same set of variables. Solving such systems involves finding values for the variables that simultaneously satisfy all equations in the system.

In the context of the investment problem, we set up a system of equations to reflect the different aspects of the investment scenario:

• Equation 1: The sum of the investments at both interest rates equals the total amount, given as \(x + y = 35000\).
• Equation 2: The sum of the interest from each investment must equal the total interest earned, expressed as \(0.085x + 0.12y = 3675\).

This system helps us link the known factors (total investment and interest) with the unknowns (amounts invested at each rate).

Solving systems of equations can be done through methods like substitution, elimination, or using matrices. For this problem, substitution proved effective because it allowed us to express one variable in terms of the other and solve straightforwardly.

Investment Distribution

Investment distribution refers to how an individual or entity divides investments among different options to achieve financial objectives.

In our example, an investor allocates \(\\(35,000\) between two funds with interest rates of \(8.5\%\) and \(12\%\). The distribution of funds is crucial to maximize returns or meet specific financial goals. Here, the investor wants to ensure the total returns match the known annual interest, \(\)3675\).

To find the distribution:

• We began by assigning variables: \(x\) for the amount invested at \(8.5\%\), and \(y\) for \(12\%\).
• The total investment equation \(x + y = 35000\) provided a direct way to express one amount in terms of another.

By calculating, we were able to determine \(x\) (\\(15,000) and \(y\) (\\)20,000), showcasing a balanced way to distribute funds to meet the investment's interest requirements.

Interest Rate

Interest rate is the percentage of the principal charged as interest by a lender to a borrower for the use of assets.

For investments, it represents the expected annual return on the principal amount invested. In this exercise, two different interest rates are given: \(8.5\%\) and \(12\%\).

These rates will affect how much return an investor can expect:

• At \(8.5\%\), the return is calculated as \(0.085 \times \text{{amount invested at this rate}}\).
• At \(12\%\), it is \(0.12 \times \text{{amount invested at this rate}}\).

The goal is to determine how much to invest at each rate so that the total interest earned is the given \($3675\). Understanding how these rates influence earnings is crucial in making informed investment decisions.