Problem 40 - College Algebra

Question

Exercises \(37-38\) involve markup, the amount added to the dealer's cost of an item to arrive at the selling price of that item. You invested \(\$ 11,000\) in two accounts paying \(5 \%\) and \(8 \%\) annual interest. If the total interest earned for the year was \(\$ 730,\) how much was invested at each rate?

Step-by-Step Solution

Answer

To solve this problem, we will use the method of substitution. The first step is to setup two equations that represent the total money invested and the total interest earned respectively. We then express one variable in terms of the other and substitute in the second equation to find one of the unknowns. After finding one, we substitute it in the first equation to find the second unknown.

Step 1: Set up the equations

We can denote by \( x \) the amount of money invested at 5% interest and by \( y \) the amount of money invested at 8% interest. We have two pieces of information that allow us to set up two equations. The first equation is based on the total investment: \( x + y = 11,000 \). The second equation is based on the total interest earned: \( 0.05x + 0.08y = 730 \).

Step 2: Solve the system of equations

There are various techniques to solve a system of linear equations, like substitution, elimination or matrix method. For simplicity, let's use substitution. From the first equation, we can express \( y \) as \( y = 11,000 - x \). Substituting \( y \) in the second equation gives us: \( 0.05x + 0.08(11,000 - x) = 730 \). Solving this equation for \( x \) will give us the amount invested at 5%.

Step 3: Find the value of the second variable

After we find the value of \( x \), we can substitute it into the equation \( y = 11,000 - x \) to find the amount of money invested at 8% interest rate.

Key Concepts

Understanding Linear Equations

In mathematics, linear equations are equations of the first degree. They involve two variables and take the form of either: \( ax + by = c \), or \( y = mx + b \). In our problem, these equations help us model financial situations, like investments. Here we have two equations:

• \( x + y = 11,000 \)
• \( 0.05x + 0.08y = 730 \)

The first equation represents the total investment, while the second represents the total interest earned. Linear equations enable us to find unknown values by expressing relationships between different quantities.

Applying the Substitution Method

The substitution method is a technique used to solve systems of linear equations. It replaces one variable with an expression from another equation. To use this method:

• Start with an equation and express one variable in terms of the other. For instance, from \( x + y = 11,000 \), express \( y \) as \( y = 11,000 - x \).
• Substitute this expression into the second equation \( 0.05x + 0.08y = 730 \).
• It becomes \( 0.05x + 0.08(11,000 - x) = 730 \), which can now be solved for \( x \).

This method simplifies finding one variable, which makes it easier to solve for the other.

Solving Investment Problems

Investment problems often involve determining how money is distributed among different accounts to meet specific financial goals. In our exercise, understanding how much is invested at different interest rates clarifies the overall profit.

• Identify different investments and their respective interest rates.
• Set up equations to represent the total amount invested and total interest earned.
• Solve these equations to determine how much money goes into each investment.

Such problems provide a realistic application of linear equations in financial planning.

Interest Calculations Explained

Interest calculations determine the profit earned from an investment over time. They help in understanding financial growth. In our problem, interest for each account is calculated using the formula:

• Interest earned = Principal amount \( \times \) Interest rate.

For the account with 5% interest, it is \( 0.05x \), and for 8% interest, it is \( 0.08y \).

• The total interest earned can then be expressed as \( 0.05x + 0.08y = 730 \).
• This equation helps us conclude how the investments balance each other to make a total interest of \( \$730 \).

Understanding these calculations is vital for making informed investment decisions.