Problem 53 - Algebra for College Students

Question

Solve each investment problem. Michelle invested some money at \(4.5 \%\) simple interest and \(\$ 1000\) less than twice this amount at \(3 \%\). Her total annual income from the interest was \(\$ 1020\). How much was invested at each rate?

Step-by-Step Solution

Answer

Michelle invested \$10000 at 4.5% and \$19000 at 3%.

Step 1: Define Variables

Let the amount Michelle invested at 4.5% be denoted as \( x \). The amount invested at 3% is \( 2x - 1000 \).

Step 2: Set Up the Interest Equations

Interest from the money invested at 4.5% is \( 0.045x \). Interest from the money invested at 3% is \( 0.03(2x - 1000) \).

Step 3: Write the Total Interest Equation

The total annual interest income is given as \$1020. Therefore, the equation is: \( 0.045x + 0.03(2x - 1000) = 1020 \).

Step 4: Simplify the Equation

Expand and simplify the equation: \( 0.045x + 0.06x - 30 = 1020 \) Combine like terms: \( 0.105x - 30 = 1020 \)

Step 5: Solve for \( x \)

Add 30 to both sides: \( 0.105x = 1050 \) Divide both sides by 0.105: \( x = 10000 \)

Step 6: Find the Amount Invested at 3%

The amount invested at 3% is: \( 2x - 1000 \) Substitute \( x \): \( 2(10000) - 1000 = 19000 \)

Key Concepts

simple interest

Simple interest is a way to calculate the interest charge on a loan or an investment. The formula for simple interest is:

\[ \text{Simple Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]

In this exercise, Michelle receives interest on her investments, where the principal amount is the money she invested, the rate is the interest rate, and the time is one year. Simple interest is straightforward since it's directly proportional to the principal amount, the interest rate, and the time period in years.

defining variables

In problems like this, defining variables helps set up equations that can be solved. Here, we define:

• \( x \): The amount invested at 4.5%.
• \( 2x - 1000 \): The amount invested at 3%, which is twice the amount invested at 4.5% minus \$1000.

By assigning these variables, you can create equations for the total interest earned.

solving equations

Solving equations involves finding the value of the unknown variables. Here's how we solve the equation from this problem:

We start with the total interest equation:\[ 0.045x + 0.03(2x - 1000) = 1020 \]
First, distribute the 3% interest rate across \(2x - 1000\):
\[ 0.045x + 0.06x - 30 = 1020 \]
Combine like terms:
\[ 0.105x - 30 = 1020 \]
Add 30 to both sides to isolate terms involving \(x\):\[ 0.105x = 1050 \]Lastly, divide both sides by 0.105 to find \(x\):\[ x = 10000 \] This means Michelle invested \$10,000 at 4.5%.

interest rates

Interest rates determine how much interest you earn on an investment. They are expressed as a percentage. In this example, two different interest rates are used:

• 4.5% for the initial amount\( x \)
• 3% for \(2x - 1000\), the amount less adjusted by \$1000

Interest rates are important because they directly impact the total interest income. The higher the rate, the more income earned.

investment amounts

Finally, discovering the exact amounts Michelle invested at each rate is essential. We already determined that \( x = 10000 \), the amount at 4.5%.

To find the amount invested at 3%, use the equation:\[ 2x - 1000 \]
Substitute \( x \) with \( 10000 \):\[ 2(10000) - 1000 = 19000 \]
Thus, Michelle invested \$19,000 at 3%, yielding an overall annual income of \$1020.