Problem 200 - Elementary Algebra

Question

Vern sold his 1964 Ford Mustang for \(\$ 55,000\) and wants to invest the money to earn him \(5.8 \%\) interest per year. He will put some of the money into Fund A that earns \(3 \%\) per year and the rest in Fund B that earns \(10 \%\) per year. How much should he invest into each fund if he wants to earn \(5.8 \%\) interest per year on the total amount?

Step-by-Step Solution

Answer

Invest approximately \$33,714.29 in Fund A and \$21,285.71 in Fund B.

Step 1: Define the Variables

Let x be the amount invested in Fund A and y be the amount invested in Fund B. Then, we know that x + y = 55,000.

Step 2: Establish the Total Interest Equation

The total interest earned from both funds should equal 5.8% of the total amount. Therefore, 0.058 * 55,000 = 3/100 * x + 10/100 * y.

Step 3: Simplify the Interest Equation

Simplify the total interest equation: 0.058 * 55,000 = 0.03x + 0.1y.

Step 4: Solve the System of Equations

We have the following system of linear equations: 1. x + y = 55,000 2. 3x + 10y = 3,190,000. To eliminate y, multiply the first equation by 10: 10x + 10y = 550,000 Then subtract the simplified interest equation from it: (10x + 10y) - (3x + 10y) = 550,000 - 3,190 7x = 236,000.

Step 5: Solve for x

Solving for x, we get: x = 236,000 / 7 x = 33,714.29 (approximately).

Step 6: Find y

Substitute x back into the first equation to find y: x + y = 55,000 33,714.29 + y = 55,000 y = 55,000 - 33,714.29 y = 21,285.71 (approximately).

Key Concepts

Linear Equations

A linear equation is an algebraic statement where each term is either a constant or the product of a constant and a single variable. Linear equations form straight lines when plotted on a graph.
In this investment problem, we have two linear equations. The first is based on the total amount Vern invests:

• x + y = 55,000

This equation tells us the sum of investments in Fund A and Fund B is $55,000.
Linear equations are the foundation for solving many algebra problems by representing real-world situations mathematically and helping us find unknowns.

Interest Calculation

Interest calculation determines how much extra money an investment will earn over time. The formula for calculating simple interest is:

• I = P * R * T

Where:

• I = Interest earned
• P = Principal amount (initial investment)
• R = Interest rate (annual)
• T = Time (years)

In the exercise, we calculate interest from two funds with different rates to match Vern's desired overall rate. The total interest equation is:

• 0.058 * 55,000 = 3/100 * x + 10/100 * y

It translates to a total interest goal of 5.8% across the combined investments.

Solving Systems of Equations

To solve systems of equations, we find values for the variables that satisfy each equation simultaneously. Here, our system consists of two key equations:

• x + y = 55,000
• 0.03x + 0.1y = 0.058 * 55,000

We look for a common solution to both equations. We can solve this system using different methods, including substitution and elimination. By eliminating one variable and solving for the other, we can find our solutions. For example,

• 10x + 10y = 550,000
• (10x + 10y) - (3x + 10y) = 550,000 - 3,190
  
  7x = 236,000

Once we find x, we substitute it back into one of our original equations to find y.

Algebraic Expressions

An algebraic expression combines numbers, variables, and arithmetic operations to represent a mathematical phrase. For investment problems like this one, expressions help formulate and solve equations for the unknowns. An expression can be simplified to make solving easier. In our exercise, we establish and simplify the interest equation:

• 0.03x + 0.1y = 0.058 * 55,000

These expressions can be manipulated by adding, subtracting, multiplying, or dividing both sides to isolate variables and find specific values relevant to the problem.