Problem 30 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Mrs. Thomas has \(\$ 15,000\) to invest. She will invest part of it in a corporate bond that pays \(8 \%\) interest per year and the rest in a stock that pays \(11 \%\) interest per year. How should she divide up the \(\$ 15,000\) so that her total yearly interest will be \(10 \%\) of her investments?

Step-by-Step Solution

Answer

Invest \( 5000 \) in the corporate bond and \( 10000 \) in the stock.

Step 1 - Define Variables

Let \( x \) be the amount of money invested in the corporate bond that pays an 8% interest. Then the remaining \( 15000 - x \) dollars is invested in the stock that pays an 11% interest.

Step 2 - Write the Interest Equations

The interest earned from the corporate bond can be written as \( 0.08x \). The interest from the stock investment can be written as \( 0.11(15000 - x) \). The total interest earned is the sum of these two amounts.

Step 3 - Set Up the Total Interest Equation

Since Mrs. Thomas wants the total yearly interest to be 10% of her investments, which is \( 0.10 \times 15000 \), we can set up the equation: \( 0.08x + 0.11(15000 - x) = 0.10 \times 15000 \).

Step 4 - Simplify the Equation

First, calculate \( 0.10 \times 15000 = 1500 \). This gives the equation: \( 0.08x + 0.11(15000 - x) = 1500 \). Now distribute the 0.11: \( 0.08x + 0.11 \times 15000 - 0.11x = 1500 \).

Step 5 - Combine Like Terms

Combine the terms involving x: \( 0.08x - 0.11x + 0.11 \times 15000 = 1500 \). Simplify to get: \( -0.03x + 1650 = 1500 \).

Step 6 - Solve for x

Subtract 1650 from both sides: \( -0.03x = 1500 - 1650 \). This gives \( -0.03x = -150 \). Now divide both sides by -0.03 to solve for x: \( x = 5000 \).

Step 7 - Calculate the Remaining Investment

Since \( x = 5000 \), the amount invested in the corporate bond is \( 5000 \). The remaining amount invested in the stock is \( 15000 - 5000 = 10000 \).

Key Concepts

Investment Distribution

In this exercise, we focus on investment distribution between two types of investments: a corporate bond and a stock. The problem states that Mrs. Thomas has \(15,000 to invest in total. She needs to find out how much to invest in each option to achieve a specific interest rate return. To distribute the investment, use one variable to represent the amount invested in one option, and the rest will be what is left from the total.

• Define the amount invested in each option.
• Express the total investment using our defined variables.

For instance, let x be the amount invested in the corporate bond. Thus, the amount left over, \)15,000 - x\(, will be invested in the stock. This ensures that the total remains \)15,000, and we can easily calculate the necessary distribution.

Interest Calculation

Interest calculation is essential in understanding how much profit Mrs. Thomas makes from her investments. Each investment option has a different interest rate.
The corporate bond offers an 8% yearly interest, written as 0.08 in decimal form. The stock offers an 11% yearly interest, which is 0.11 in decimal form.

• Calculate the interest from the corporate bond: 0.08x.
• Calculate the interest from the stock: 0.11(15000 - x).

The total interest from both investments is the sum of these interests. As per the problem, it needs to be 10% of the total investment (\(15,000). This means we need our equation to show the total interest equaling 0.10 \times 15000 = \)1500. Combining all these into an equation helps us reach that specific total interest goal.

Solving Linear Equations

The core of solving this problem lies in setting up and solving a linear equation. Let's simplify the previously formulated interest equation and solve for the variable. We start with 0.08x + 0.11(15000 - x) = 1500.

• Distribute 0.11 in our equation: 0.08x + 0.11 \times 15000 - 0.11x = 1500.
• Combine like terms involving x: 0.08x - 0.11x + 1650 = 1500.

This simplification gives us a more manageable equation: -0.03x + 1650 = 1500.
Next, solve for x:

• Subtract 1650 from both sides: -0.03x = -150.
• Divide by -0.03: x = 5000.

Therefore, Mrs. Thomas should invest \(5000 in the corporate bond. The remaining amount, \)10,000, will be invested in the stock. This distribution will achieve her goal of a 10% total yearly interest.