Problem 52 - A Graphical Approach to College Algebra

Question

A student won \(\$ 100,000\) in the Louisiana state lottery. He invested part of the money in real estate with an annual return of \(5 \%\) and another part in a money market account at \(0.5 \%\) interest. He invested the rest, which amounted to \(\$ 20,000\) less than the sum of the other two parts, in certificates of deposit that pay \(1.75 \%\) If the total annual interest on the money was \(\$ 3250,\) how much was invested at each rate?

Step-by-Step Solution

Answer

Real estate: \$53,333.33, Money market: \$6,666.67, Certificates: \$40,000.

Step 1: Define Variables

Let \( x \) be the amount invested in real estate (5%) and \( y \) be the amount invested in the money market (0.5%). The third part invested in certificates of deposit (1.75%) would thus be \( x + y - 20,000 \).

Step 2: Set Up Equations

Since the total investment was \$100,000, we have:1. \( x + y + (x + y - 20,000) = 100,000 \) 2. The total interest equation is: \( 0.05x + 0.005y + 0.0175(x + y - 20,000) = 3250 \).

Step 3: Simplify the Equations

From Equation 1: \( 2x + 2y = 120,000 \), simplifying, we get \( x + y = 60,000 \).From Equation 2, simplify it to get: \( 0.0675x + 0.0225y = 3750 \).

Step 4: Solve Simultaneous Equations

Solve \( x + y = 60,000 \) and \( 0.0675x + 0.0225y = 3750 \). Substituting \( y = 60,000 - x \) into the second equation:\( 0.0675x + 0.0225(60,000 - x) = 3750 \).Simplify and solve for \( x \):\( 0.0675x + 1350 - 0.0225x = 3750 \)\( 0.045x = 2400 \)\( x = 53,333.33 \).

Step 5: Find Values for all Parts

Substitute \( x = 53,333.33 \) back to find \( y \):\( y = 60,000 - 53,333.33 = 6,666.67 \).Then, the amount in certificates of deposit is:\( 53,333.33 + 6,666.67 - 20,000 = 40,000 \).

Key Concepts

Investment Distribution

In the world of finance, knowing how to wisely allocate funds can significantly impact returns. For instance, if you have \( \\(100,000 \) and need to distribute it across various investment avenues, strategic planning is crucial.
In our example, the student splits the money into three distinct parts:

• Investing some in real estate with an annual return of 5%.
• Allocating some to a money market account at 0.5% interest.
• Putting the rest into certificates of deposit offering 1.75% interest.

The challenge arises when you consider the condition that one part of the money should be \( \\)20,000 \) less than the combined total of the other two investments. This condition not only influences how much is allocated to each part but also highlights the necessity of balancing risk and return among various options.

Simultaneous Equations

To tackle the investment problem, simultaneous equations come into play, allowing us to calculate the precise amounts allocated to each investment. We establish variables for the unknowns: let \( x \) represent the amount invested in real estate, and \( y \) for the money market account.
The sum of investments across all these assets equals \(100,000, leading to the equation:

• \( x + y + (x + y - 20,000) = 100,000 \)

By simplifying, we derive: \( x + y = 60,000 \). Subsequently, for the total interest of \( \\)3250 \), we set up another equation:

• \( 0.05x + 0.005y + 0.0175(x + y - 20,000) = 3250 \)

Breaking these down into more manageable forms helps us arrive at the correct investment amounts for each type.
Solving such equations allows us to simultaneously satisfy multiple conditions, streamlining the solution process.

Interest Calculation

Interest calculation is pivotal in determining how much return is generated from each investment. Each type of investment in our scenario offers distinct interest rates:

• Real estate at 5% annually
• Money market account at 0.5% per annum
• Certificates of deposit yielding 1.75% annually

To find out how much interest each investment earns, we multiply the principal amount invested by their respective interest rate.
For instance, the formula to calculate interest for the real estate portion is \( 0.05x \). Similarly, interest for the money market is \( 0.005y \), and for the certificates of deposit, it is \( 0.0175(x + y - 20,000) \).
The sum of all these individual interests should match the total interest earned, given by the problem statement: \( \$3250 \). This process ensures each investment is contributing accurately to the overall return.