Problem 29 - 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 \(\$ 400\) Rate 1 \(4 \%\) Rate 2 \(2 \%\)

Step-by-Step Solution

Answer

The amounts of money invested in fund 1 and fund 2 are \$10,000 and \$8000 respectively.

Step 1: Formulate Equations

Formulate two equations based on the problem.The first equation is based on the total sum of money distributed in the two funds:x + y = \$18,000The second equation is based on how the total interest is generated. 4% of x (Rate 1's return) and 2% of y (Rate 2's return) total up to the given yearly interest:0.04x + 0.02y = \$400

Step 2: Solve the First Equation for x

Isolate x in the first equation to be able to substitute it in the second equation:x = \$18,000 - y

Step 3: Substitute x in the Second Equation

Replace x in the second equation with the value found above:0.04(\$18,000 - y) + 0.02y = \$400

Step 4: Solve the Equation for y

By simplifying the equation obtained in step 3, we compute the value for y:0.04 * \$18,000 - 0.04y + 0.02y = \$400y = \$8000

Step 5: Solve the Equation for x

Substitute the found value of y in the first equation to determine the value of x:x = \$18,000 - \$8000x = \$10,000

Key Concepts

Simple Interest

Simple interest is a way to calculate the interest you earn or pay on a certain amount of money, based on an interest rate. It is called "simple" because it doesn’t involve compounding, where interest can earn its own interest over time.

Here's how you calculate it: If you invest an amount of money, called the principal, at a fixed interest rate for a certain period, your interest is simply a product of these three things:

• Principal (the original amount of money)
• Rate of Interest (usually an annual percentage)
• Time (number of years, if the interest rate is per year)

The formula for simple interest is: \[ I = P \times r \times t \]Where:

• \( I \) is the interest
• \( P \) is the principal
• \( r \) is the rate of interest (expressed as a decimal)
• \( t \) is the time

In the problem, we see a total interest of \$400 earned from investments in two funds with simple interest rates of 4% and 2% respectively.

Investment Allocation

Investment allocation is how you distribute your money among different investment options. This strategy is crucial to managing risks and maximizing returns based on different rates and conditions.

Consider two funds in our example: One fund provides a 4% interest rate, and the other offers a 2% rate. When investing a total amount, it’s important to decide how much to put into each option. These decisions affect the total earning, as seen in the total interest received.

• If you invest more in a high-rate fund, you can earn more interest.
• More in a low-rate fund might mean less interest, but potentially less risk, too.

In the exercise, we have \\(18,000 allocated between two funds. Solving an investment allocation problem helps determine how much goes into each fund to meet specific financial goals, like earning exactly \\)400 in interest annually.

Linear Equations

Linear equations are mathematical expressions that express relationships between variables using constants (fixed values) and coefficients (numerical factors). They are called "linear" because they graph as straight lines.

In our case, we deal with two variables reflecting the amounts invested in two funds. These equations help us understand and solve real-world finance issues, such as investment distribution across different interest rates. The goal is to find the values of these variables to satisfy all conditions given in the problem:

• First equation represents the total amount invested: \( x + y = 18,000 \).
• Second equation combines the interests from both funds: \( 0.04x + 0.02y = 400 \).

Solving this system of equations involves using substitution or elimination methods to find the amounts \( x \) and \( y \). Understanding these concepts makes it easier to handle various financial scenarios effectively.