Problem 72 - College Algebra

Question

Use a system of linear equations with two variables and two equations to solve. If an investor invests a total of $$\$ 25,000$$ into two bonds, one that pays \(3 \%\) simple interest, and the other that pays \(2 \frac{7}{8} \%\) interest, and the investor earns $$\$ 737.50$$ annual interest, how much was invested in each account?

Step-by-Step Solution

Answer

$15,000 invested at 3% and $10,000 at 2.875%.

Step 1: Define Variables

Let \( x \) be the amount invested in the bond that pays 3% interest, and \( y \) be the amount invested in the bond that pays 2.875% (which is equivalent to \(2 \frac{7}{8} \%\)).

Step 2: Create Linear Equations

We need to form two equations based on the information given: 1. The total investment equation: \[ x + y = 25000 \] 2. The total interest equation:\[ 0.03x + 0.02875y = 737.50 \]

Step 3: Solve for One Variable

From the first equation, we can express \( x \) in terms of \( y \): \[ x = 25000 - y \]

Step 4: Substitute and Solve

Substitute \( x = 25000 - y \) back into the second equation: \[ 0.03(25000 - y) + 0.02875y = 737.50 \]Simplify this equation:\[ 750 - 0.03y + 0.02875y = 737.50 \]\[ 750 - 0.00125y = 737.50 \]Subtract 750 from both sides:\[ -0.00125y = -12.5 \]Solve for \( y \):\[ y = \frac{-12.5}{-0.00125} \]\[ y = 10000 \]

Step 5: Solve for Other Variable

Now that we have \( y = 10000 \), substitute back to find \( x \):\[ x = 25000 - 10000 \]\[ x = 15000 \]

Step 6: Verification Step: Check Total Interest

Verify by calculating the total interest:Interest from \( x \): \( 0.03 \times 15000 = 450 \)Interest from \( y \): \( 0.02875 \times 10000 = 287.5 \)Total Interest: \( 450 + 287.5 = 737.5 \). The total interest is correct.

Key Concepts

Simple Interest

Simple interest is a method to calculate the interest charge on a loan or investment. It's straightforward and easy to understand. Here's how it works:
When you invest money or take a loan, the interest is calculated only on the principal amount. That means the initial sum of money you invest or borrow remains unchanged during the interest calculation.

• The key formula is: \( \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \)
• "Principal" is the initial amount of money.
• "Rate" is the percentage of interest per time period.
• "Time" is the time period the money is invested or borrowed for.

In the given exercise, the investor is earning simple interest from two different bonds. The easy part is you don't have to calculate the interest on the interest earned, just the original amount.

Investment

Investment pertains to the allocation of money in hopes of generating more money over time. Real-life examples include buying stocks, bonds, or real estate.
In this exercise, we focus on investing in bonds that pay periodic interest. Here's why people choose bonds:

• Bonds are often seen as a safer investment compared to stocks because they provide fixed interest over time.
• A bond represents a loan to a company or government, and in exchange, they pay you interest until the bond matures.

The concept at play here is diversifying investment funds between bonds that pay different interest rates, which allows maximizing potential returns while balancing risk.

Linear Equation Solution

A linear equation is a mathematical statement that expresses the relationship between variables in a straight line when plotted on a graph. Solving linear equations often involves simplifying equations to find the values of the variables.
In the exercise, two linear equations form a system that can be solved:

• First equation involves the total investment: \( x + y = 25000 \)
• Second equation involves the total interest earned: \( 0.03x + 0.02875y = 737.50 \)
• By substituting one equation into the other, we simplify and solve for one variable at a time.

This is an efficient way to solve real-world problems using algebraic methods built on logical steps.

Variable Definition

Defining variables is an essential step in forming equations and solving mathematical problems. A variable represents an unknown quantity we aim to find.
In tackling the given exercise:

• \( x \) was chosen to represent the amount invested in the bond with a 3% interest rate.
• \( y \) was used for the amount invested in the bond with a 2.875% interest rate.

By clearly defining variables, we can set up equations that accurately reflect the problem's conditions, making it easier for us to identify and plug in values. This step simplifies the process of arriving at the solution and reduces errors along the way.