Problem 26 - Precalculus: Functions and Graphs

Question

A woman has \(\$ 19,000\) to invest in two funds that pay simple interest at the rates of \(4 \%\) and \(6 \%\) per year. Interest on the \(4 \%\) fund is tax- exempt; however, income tax must be paid on interest on the 6\% fund. Being in a high tax bracket, the woman does not wish to invest the entire sum in the 6\% account. Is there a way of investing the money so that she will receive \(\$ 1000\) in interest at the end of one year?

Step-by-Step Solution

Answer

Invest \$7000 in the 4% fund and \$12000 in the 6% fund.

Step 1: Define Variables

Let \( x \) be the amount invested in the 4% fund and \( y \) be the amount invested in the 6% fund. We know that \( x + y = 19000 \).

Step 2: Express Total Interest

The interest from the 4% investment is \( 0.04x \) and from the 6% investment is \( 0.06y \). The total interest desired is \$1000, which leads to the equation: \( 0.04x + 0.06y = 1000 \).

Step 3: Solve the System of Equations

From the two equations: \( x + y = 19000 \) and \( 0.04x + 0.06y = 1000 \), we can substitute \( y = 19000 - x \) into the second equation and solve for \( x \).

Step 4: Substitute and Simplify

Substituting \( y = 19000 - x \) into \( 0.04x + 0.06y = 1000 \), we get:\[ 0.04x + 0.06(19000 - x) = 1000 \].Simplifying, \( 0.04x + 1140 - 0.06x = 1000 \) leads to \( -0.02x + 1140 = 1000 \).

Step 5: Find the Value of x

From \( -0.02x + 1140 = 1000 \), simplify to find \( x \):\[ -0.02x = 1000 - 1140 \]\[ -0.02x = -140 \]\[ x = 7000 \].

Step 6: Find the Value of y

Substitute \( x = 7000 \) back into \( x + y = 19000 \) to find \( y \):\[ 7000 + y = 19000 \]\[ y = 12000 \].

Step 7: Verify the Solution

Check whether the interest from \( x = 7000 \) and \( y = 12000 \) equals \$1000.The interest from \( x \) is \( 0.04 \times 7000 = 280 \) and from \( y \) is \( 0.06 \times 12000 = 720 \), giving a total interest of \( 280 + 720 = 1000 \). This confirms the solution is correct.

Key Concepts

Investment Strategy

Investing is like planning a financial journey. It's not just about choosing any destination, but about determining how and where you want to allocate your resources to achieve your goals. In this scenario, the woman has a choice between two funds offering different interest rates. One fund offers a safe 4% return which is tax-exempt, while the other offers a higher 6% return but incurs taxes.

Her strategy involves balancing these options to maximize her earnings while ensuring tax efficiency. Since the higher interest is taxable, she needs to be cautious about over-investing in this fund. By carefully diversifying her investment between the two funds, she can optimize her earnings and meet her target of earning $1,000 in interest by the year’s end.

Key steps in investment strategy include:

• Identifying the available options and their benefits.
• Considering the impact of taxes on investment returns.
• Balancing risk and reward to achieve financial objectives.

System of Equations

To tackle investment decisions analytically, we can use a system of equations. This method allows us to define variables for the amounts in each fund and then create equations to represent the financial and interest constraints.

Here, we introduced two equations to solve the problem:

• \( x + y = 19000 \) - represents the total sum available to invest.
• \( 0.04x + 0.06y = 1000 \) - represents the desired total interest after one year.

Solving these equations together helps in determining the optimal amounts to invest in each fund.

Working with systems of equations provides a clear structure to balance multiple constraints and goals simultaneously. This method is particularly useful when dealing with financial decisions involving multiple variables and outcomes.

Tax Implications

Taxes can significantly affect the overall return on an investment, making it essential to understand their impact. In this situation, the two investment options have different tax implications. The 4% fund is tax-exempt, meaning investors keep all the earned interest, while the 6% fund requires payment of taxes on the interest earned.

When investing, it's crucial to consider how taxes will diminish returns, especially for high-income individuals in a higher tax bracket. By minimizing taxable investments, the investor can retain more of their earnings. It involves:

• Selecting tax-advantaged investments to maximize returns.
• Calculating the after-tax return to better evaluate an investment's profitability.

Managing tax implications well is an integral part of a sound investment strategy, allowing investors to enhance their earnings effectively.

Algebraic Substitution

Algebraic substitution is a key technique used to solve the system of equations in this exercise. It involves expressing one variable in terms of another, allowing us to substitute and solve equations step-by-step.

Here, we used the substitution method to find the values of \( x \) and \( y \):

• First, solve for one variable: \( y = 19000 - x \).
• Substitute this expression back into the second equation: \( 0.04x + 0.06(19000 - x) = 1000 \).
• Simplify and solve for \( x \).

Substitution breaks down a complex problem into manageable steps, making it easier to understand and solve. This method is particularly helpful in investment scenarios where multiple conditions must be met simultaneously.