Problem 54 - Algebra for College Students

Question

Solve each investment problem. Plotr invested some money at \(3.5 \%\) simple interest, and \(\$ 5000\) more than three times this amount at \(4 \%\). He earned \(\$ 1440\) in annual interest. How much did he invest at each rate?

Step-by-Step Solution

Answer

Plotr invested \( 8000 \) at \( 3.5\% \) and \( 29000 \) at \( 4\% \).

Step 1: Define Variables

Let \( x \) be the amount invested at \( 3.5\% \) interest. Therefore, the amount invested at \( 4\% \) is \( 3x + 5000 \).

Step 2: Write Interest Formulas

The interest earned from \( x \) investment at \( 3.5\% \) is \( 0.035x \). The interest earned from \( 3x + 5000 \) investment at \( 4\% \) is \( 0.04(3x + 5000) \).

Step 3: Write the Total Interest Equation

The total interest earned is given as \( 1440 \). So, the equation is \( 0.035x + 0.04(3x + 5000) = 1440 \).

Step 4: Simplify the Equation

Combine like terms to simplify the equation: \( 0.035x + 0.12x + 200 = 1440 \).

Step 5: Solve for \( x \)

Combine the terms involving \( x \): \( 0.155x + 200 = 1440 \). Subtract \( 200 \) from both sides: \( 0.155x = 1240 \). Divide by \( 0.155 \): \( x = 8000 \).

Step 6: Find the Amount Invested at 4\%

The amount invested at \( 4\% \) is \( 3x + 5000 \). Using \( x = 8000 \), we get \( 3(8000) + 5000 = 24000 + 5000 = 29000 \).

Key Concepts

Simple Interest

Understanding simple interest is the foundation for solving many investment problems. Simple interest is calculated on the original amount of money, which is called the principal. It is calculated using the formula:
\[{ I = P \times R \times T }\] where:
- \( I \) = Interest earned
- \( P \) = Principal amount invested
- \( R \) = Rate of interest per period
- \( T \) = Time period

For example, if we invest \( x \) dollars at a rate of \( 3.5\% \), the interest earned in one year would be \( 0.035x \). Similarly, investing \( 3x + 5000 \) dollars at \( 4\% \) would yield an interest of \( 0.04(3x + 5000) \). Using the simple interest formula helps us set up the problem correctly.

Solving Linear Equations

Solving linear equations is a methodical process often used in investment problems to find unknown values. The goal is to isolate the variable on one side of the equation. Let's break down the steps:

1. **Set up the Equation**: First, write down the equation based on the problem statement. Here we have: \[{ 0.035x + 0.04(3x + 5000) = 1440 }\]
2. **Distribute and Combine**: Distribute the terms and combine like terms: \[{ 0.035x + 0.12x + 200 = 1440 }\]
3. **Isolate the Variable**: Move all constants to the other side: \[{ 0.155x + 200 = 1440 }\]
\[{ 0.155x = 1240 }\]
4. **Solve for the Variable**: Divide both sides by the coefficient of x: \[{ x = \frac{1240}{0.155} \rightarrow x = 8000 }\]

By following these steps, we found that the initial investment at \( 3.5\% \) was \( 8000 \) dollars.

Investment Calculations

Investment calculations involve using the information and derived equations to find how much money was invested. Here's how we apply this concept:

1. **Define Investment Amounts**: Let's say \( x \) represents the amount invested at \( 3.5\% \). The problem states that the amount invested at \( 4\% \) is \( 3 \) times \( x \) plus an additional \( 5000 \) dollars.
2. **Calculate Individually**: From solving the linear equation, we found \( x = 8000 \). Hence, the amount at \( 4\% \) is calculated as \( 3 \times 8000 + 5000 \):
\[{ 3(8000) + 5000 = 29000 }\]
3. **Check Total Investment**: Adding both investments, we get:
\[{ 8000 + 29000 = 37000 }\]
4. **Verify Interest Earned**: To ensure correctness, we calculate the interest earned:
\[{ 0.035 \times 8000 = 280 }\]
\[{ 0.04 \times 29000 = 1160 }\]
\[{ 280 + 1160 = 1440 }\]

By calculating these values, we confirm the accuracy of our solution. Plotr's investments and interests match the given problem statement.