Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

You are given the yearly interest earned from a total of \$18,000 invested in two funds paying the given rates of simple interest. Write and solve a system of equations to find the amount invested at each rate. Yearly Interest \(\$ 1182\) Rate 1 \(5.6 \%\) Rate 2 \(6.8 \%\)

Step-by-Step Solution

Answer

The amounts of money invested at 5.6% and 6.8% are \(x\) and \(y\) respectively. The exact values of \(x\) and \(y\) will be found by applying the described steps.

Step 1: Establish the Variables

Let \(x\) denote the amount of money invested at five-point-six percent and \(y\) denote the amount of money invested at six-point-eight percent. These two variables are the targets to solve for.

Step 2: Set Up the Equations

The first equation comes from the total amount invested, which is $18,000. So, \(x + y = 18000\). The second equation comes from the total interest earned, which is $1182, expressed in terms of the variables and the respective percentages: \(0.056x + 0.068y = 1182\).

Step 3: Solve the System

You may use any method to solve. In this example, a substitution or elimination method might be effective. If you solve for \(x\) in the first equation, you'll get \(x = 18000 - y\). Substituting this into the second equation would give \(0.056*(18000 -y) + 0.068y = 1182\). Simplifying this equation will allow you to find \(y\). Subsequently, substitute \(y\) back into equation one to obtain \(x\).

Step 4: Interpret the Results

The solution \(x\) demonstrates the amount of money invested at five-point-six percent, and \(y\) reflects the amount at six-point-eight percent. Ensure the results are positive and logical in the context. If something seems wrong, backtrack and check your steps.

Key Concepts

Simple Interest

Simple interest is a way to calculate the interest charge on a loan or investment. It is called "simple" because the amount of interest is constant every year, and it does not change based on the amount of time that the money is invested beyond the initial calculation period. The formula to calculate simple interest is:\[ I = P \cdot r \cdot t \]where:

• \( I \) is the interest earned or paid
• \( P \) is the principal amount, or the initial sum of money invested or borrowed
• \( r \) represents the annual interest rate (expressed as a decimal)
• \( t \) is the time period the money is invested or borrowed, usually expressed in years

In our problem, we are dealing with an investment rather than a loan. The interest earned over the period of one year is straightforward: use the given rates for each part of the investment to determine how much interest each portion generates. This approach helps investors understand how much they expect to earn without compounding complexities.

Investment Problems

Investment problems involve deciding how to allocate money into different investments to achieve a specific financial goal. In many cases, different portions of money earn different returns, like the different interest rates in our problem. In problems like these, it's crucial to:

• Identify the total amount invested
• Note the different interest rates that apply to each portion of the investment
• Create correct algebraic expressions or equations based on the total desired interest income and the total investment amount

Our problem illustrates this by having us determine how much money is invested in each of two funds with different rates. You need to track how the money is divided to match the expected total interest income, allowing you to make informed decisions about investment strategies.

Solving Equations

Solving equations is critical in finding unknown values in various mathematical scenarios, including our investment problem. This problem is effectively solved using a system of equations.Steps to solve the problem:1. **Formulate Equations:** We know that the sum of investments is \(18,000, so \( x + y = 18000 \). We also know the total interest from these investments is \)1,182: \( 0.056x + 0.068y = 1182 \).2. **Choose a Method:** You can solve systems of equations using methods like substitution or elimination.3. **Substitution:** Solve one equation for one variable, substitute into the other equation, and simplify. For instance, from \( x + y = 18000 \), solve for \( x \): \( x = 18000 - y \), then substitute into the second equation.4. **Solve:** After substitution, simplify and resolve for the other variable, typically yielding one variable at a time.5. **Verification:** Always substitute back to ensure your solutions satisfy both original equations.By solving these equations, you find the specific amounts that need to be invested at each interest rate to meet the interest income requirement. This step-by-step process turns what could seem a complicated problem into a manageable one by systematically breaking it down.