Problem 69 - College Algebra and Calculus: An Applied Approach

Question

Investment Mix You invest \(\$ 15,000\) in two funds paying \(6.5 \%\) and \(7.5 \%\) simple interest. The total annual interest is \(\$ 1020 .\) How much do you invest in each fund?

Step-by-Step Solution

Answer

Therefore, $12,500 is invested in the 6.5% fund and $2,500 is invested in the 7.5% fund.

Step 1: Establish Variables

Let \( x \) represent the investment in the fund with 6.5% interest and \( y \) represent the investment in the fund with 7.5% interest.

Step 2: Set Up the Equations

We know that the total amount invested is $15,000, so our first equation is: \( x + y = 15000 \). We also know that the total interest earned is $1020, which gives us the second equation: \( 0.065x + 0.075y = 1020 \).

Step 3: Solve the System of Equations

Solving simultaneously the two equations, first multiply the first equation by 0.065 and subtract it from the second equation in order to eliminate x: \( 0.075y - 0.065y = 1020 - 0.065*15000 \) implies \( 0.01y = 25 \) (1) will give \( y = 2500 \). Substituting \( y = 2500 \) into the first equation to find x: \( x = 15000 - 2500 = 12500 \).

Key Concepts

Simple Interest

Simple interest is a fundamental concept in finance, used to calculate the interest on an investment or loan. It is called "simple" because the interest is calculated only on the principal amount, or the initial sum of money put into the investment. The formula for simple interest is:
\[ I = P \times r \times t \]where:

• \( I \) is the interest earned or paid,
• \( P \) is the principal amount,
• \( r \) is the annual interest rate (as a decimal), and
• \( t \) is the time in years.

In the exercise, simple interest is applied to two different funds with different rates. The total interest from both funds combined is given as $1020, helping form one of the equations needed to solve the problem.

System of Equations

When dealing with more than one unknown, systems of equations come into play. In this exercise, there are two unknowns: the amounts invested in each fund. We have two pieces of information:

• The total amount invested is \(15,000.
• The total annual interest earned is \)1020.

These facts translate into a system of two equations:
1. \( x + y = 15000 \), where \( x \) and \( y \) are the investments in the 6.5% and 7.5% funds respectively.
2. \( 0.065x + 0.075y = 1020 \), representing the interest from both funds.
By having two equations, you can solve for the two variables using methods such as substitution or elimination. This provides a complete picture of how the total investment is split.

Algebraic Solutions

Solving a system of equations algebraically involves manipulating the equations to find the values of the unknowns. For this exercise:Through the elimination method, you simplify one variable out of the equations. First, you adjust the equations:
Step 1: Multiply the first equation by 0.065 to balance the coefficients: \( 0.065x + 0.065y = 0.065 \times 15000 \).
Step 2: Subtract this from the second equation to eliminate \( x \):

• \( 0.075y - 0.065y = 1020 - 0.065 \times 15000 \)

This leads to \( 0.01y = 25 \), thus giving us \( y = 2500 \).
With \( y \) known, substitute back into the first equation to find \( x \):

• \( x = 15000 - 2500 = 12500 \)

This algebraic method ensures a precise solution, unraveling how the initial amount is distributed among two funds based on their contributions to the total interest.