Problem 199 - Elementary Algebra

Question

Vartan was paid \(\$ 25,000\) for a cell phone app that he wrote and wants to invest it to save for his son's education. He wants to put some of the money into a bond that pays \(4 \%\) annual interest and the rest into stocks that pay \(9 \%\) annual interest. If he wants to earn \(7.4 \%\) annual interest on the total amount, how much money should he invest in each account?

Step-by-Step Solution

Answer

Invest \( 8000 \) dollars in bonds and \( 17000 \) dollars in stocks.

Step 1: Define Variables

Let the amount invested in bonds be denoted by \( x \). Then the amount invested in stocks will be \( 25000 - x \).

Step 2: Formulate Equations

Vartan wants an average annual interest of \( 7.4 \% \). Set up the equation based on the interest from both investments: \( 0.04x + 0.09(25000 - x) = 0.074 \times 25000 \).

Step 3: Simplify the Equation

Distribute and simplify the equation: \( 0.04x + 2250 - 0.09x = 1850 \).

Step 4: Combine Like Terms

Combine the \( x \) terms: \( 2250 - 0.05x = 1850 \).

Step 5: Isolate \( x \)

Subtract \( 2250 \) from both sides: \( -0.05x = -400 \).

Step 6: Solve for \( x \)

Divide both sides by \( -0.05 \): \( x = 8000 \).

Step 7: Determine Other Investment

Since \( x = 8000 \), the amount invested in bonds is \( 8000 \) dollars. The amount invested in stocks is \( 25000 - 8000 = 17000 \) dollars.

Key Concepts

Understanding Simple Interest

Simple interest is a common way to calculate interest earnings on an investment. It is straightforward: the interest earned is a fixed percentage of the principal amount over a certain period of time. The formula for simple interest is:

• \[ \text{Simple Interest (SI)} = \text{Principal} \times \text{Rate} \times \text{Time} \]

where Principal is the initial amount invested, Rate is the annual interest rate, and Time is in years. For instance, if Vartan invests \$8000 in a bond with a \(4\%\) interest rate, the interest for one year would be \[ 8000 \times 0.04 \times 1 = 320 \text{ dollars} \].
Simple interest is handy for quick calculations and is often used in bonds or savings accounts. It does not compound, making it easier to predict earnings.

Algebraic Equations and Investment

To solve investment problems, we often rely on algebraic equations. These equations help us balance the different components of an investment. For Vartan's problem, we first define variables:

• Let \( x \) represent the amount invested in bonds.
• The remaining amount, \( 25000 - x \), is invested in stocks.

Next, we establish an equation based on expected returns:

• \[ 0.04x + 0.09(25000 - x) = 0.074 \times 25000 \]

Here, \( 0.04 \) and \( 0.09 \) represent the interest rates for bonds and stocks respectively, and \( 0.074 \) is the desired average return.
Then, we simplify the equation through distribution and combination of like terms, isolating \( x \) to find the exact investment amounts.
This process showcases how algebraic equations are essential tools for financial planning and investment decisions.

Working with Percentages

Percentages play a critical role in understanding interest rates and investment growth. They help convey the rate of return on various investments. For example:

• A \(4\%\) annual interest on bonds means earning 4 dollars for every 100 dollars invested per year.
• Similarly, a \(9\%\) return on stocks means earning 9 dollars per 100 dollars invested annually.

Vartan desires a weighted average return of \(7.4\%\) on his \$25000 investment. This average is calculated by the equation:
\[ \text{Total Interest Earned} = \text{Interest from Bonds} + \text{Interest from Stocks} \]
Converting this idea into percentages, we ensure the returns from the bond and stock investments meet his financial goals.
Percentages are a straightforward way to compare and decide among different investment options, revealing their potential returns clearly.

Effective Financial Planning

Financial planning involves making informed decisions on managing and investing money to meet specific goals, such as saving for education or retirement. For Vartan:

• He had \$25000 and wanted a balanced investment strategy.
• By splitting his money between bonds and stocks, he could achieve a stable and higher average return.

He needed to consider risks and rewards:

• Bonds provided a safer, low-risk return of \(4\%\).
• Stocks offered a higher return of \(9\%\), but with more risk.

By solving the algebraic equation, Vartan invested \$8000 in bonds and \$17000 in stocks to reach his desired \(7.4\%\) overall return.
This example underlines the importance of strategic investment allocation in effective financial planning.
Understanding simple interest, algebra, and percentages enhances one's ability to create and execute sound financial plans.