Problem 28 - Algebra and Trigonometry with Analytic Geometry

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 \$12,000 in the 6% fund.

Step 1: Define Variables

Let's define two variables: let \( x \) be the amount invested in the \(4\%\) fund, and \( y \) be the amount invested in the \(6\%\) fund. The total amount invested should be \$19,000. Thus, \( x + y = 19,000 \).

Step 2: Write Interest Equation

The interest from the \(4\%\) fund is \( 0.04x \), and the interest from the \(6\%\) fund is \( 0.06y \). We want the total interest to be \$1000. Therefore, the equation for the total interest is: \( 0.04x + 0.06y = 1000 \).

Step 3: Solve System of Equations

We have the system of equations: \( x + y = 19,000 \) and \( 0.04x + 0.06y = 1000 \). Substitute \( y = 19,000 - x \) into the interest equation: \( 0.04x + 0.06(19,000 - x) = 1000 \).

Step 4: Simplify and Solve for x

Substitute to get: \( 0.04x + 1,140 - 0.06x = 1000 \). Simplify to find \( -0.02x = -140 \) and solve for \( x \): \( x = 7,000 \).

Step 5: Solve for y

Using \( y = 19,000 - x \), substitute \( x = 7,000 \) to find \( y \): \( y = 19,000 - 7,000 = 12,000 \).

Step 6: Verify solution

Check the interest equation with \( x = 7,000 \) and \( y = 12,000 \): \( 0.04(7,000) + 0.06(12,000) = 280 + 720 = 1000 \), which matches the required interest of \$1000.

Key Concepts

Simple Interest

Simple interest is an easy way to calculate the interest earned or paid on an investment or loan. It is straightforward because you only need to consider the principal amount, interest rate, and time period of the investment or loan. Unlike compound interest, simple interest does not involve interest on interest.To calculate simple interest, use the formula:\[I = P \times r \times t\]Where:

• \( I \) is the interest earned or paid.
• \( P \) is the principal, which is the initial amount of money invested or borrowed.
• \( r \) is the annual interest rate expressed as a decimal.
• \( t \) is the time in years.

Let's consider an example where you have invested \\(1,000 at an interest rate of \(5\%\) for 2 years. The simple interest would be:\[I = 1,000 \times 0.05 \times 2 = \\)100\]This method keeps calculations manageable and transparent, making it easy for investors or borrowers to know how much they will earn or owe at any given point.

System of Equations

A system of equations is a set of two or more equations that involve the same set of variables. Solving such a system involves finding values for the variables that satisfy all the equations in the system.In the context of investment math problems, particularly when dealing with multi-variable situations such as our example with two different investment funds, systems of equations are vital.For example, in our investment problem, we had:

• The total amount equation: \( x + y = 19,000 \)
• The interest equation: \( 0.04x + 0.06y = 1,000 \)

To solve this system, substitution or elimination methods are typically used. For instance, by expressing one variable in terms of another using the substitution method, we can aid in eliminating one variable to solve the other straightforwardly.The strategic approach of working with systems of equations lets you tackle complex financial arrangements by breaking them down into manageable mathematical problems.

Interest Rates

Interest rates are percentages that represent the cost of borrowing money or the return on investment. They are fundamental in both personal finance and wider economic activities.Interest rates can be simple or compound. In this particular problem, we have "simple interest rates."Consider:

• A simple interest rate gives a linear growth on the original principal amount.
• They are often used for loans and investments with short durations or specific requirements, such as tax-exempt scenarios.

Understanding interest rates is essential, as they impact how much you earn from an investment or how much you must repay on a loan. For example, even a small difference can lead to substantially different outcomes when determining how much interest accumulates over time.Smart financial planning hinges on knowing what interest rates mean for your money, and in this problem, distinguishing between the \(4\%\) and \(6\%\) rates forms the basis of finding the optimal investment strategy.

Tax-exempt Investment

A tax-exempt investment is one where the interest or income generated is not subject to income tax. These investment opportunities are particularly appealing to individuals in high tax brackets because they can help maximize returns by reducing tax liabilities.In the given problem:

• The \(4\%\) fund is tax-exempt, meaning any interest earned on the amount invested here will not be taxed.
• This makes it advantageous for investors who wish to retain more of their earnings without the impact of taxation.

The decision to invest in a tax-exempt account generally depends on several factors, including the investor’s broader financial situation, tax liabilities, and investment goals. However, these opportunities often have specific stipulations or limits.Making use of tax-exempt investments can be a savvy move for protecting one's earnings from being eroded by taxes, though it requires careful consideration of one's overall financial strategy.