Problem 201 - Elementary Algebra

Question

Stephanie inherited \(\$ 40,000\). She wants to put some of the money in a certificate of deposit that pays \(2.1 \%\) interest per year and the rest in a mutual fund account that pays \(6.5 \%\) per year. How much should she invest in each account if she wants to earn \(5.4 \%\) interest per year on the total amount?

Step-by-Step Solution

Answer

Stephanie should invest \$10,000 in the CD and \$30,000 in the mutual fund.

Step 1: Define Variables

Let the amount of money invested in the certificate of deposit (CD) be denoted as \( x \). Therefore, the amount invested in the mutual fund account will be \( 40,000 - x \).

Step 2: Set Up the Interest Equations

The interest earned from the CD after one year is \( 0.021x \). The interest earned from the mutual fund account after one year is \( 0.065(40,000 - x) \).

Step 3: Establish the Total Interest Equation

Stephanie wants to earn a total of \( 5.4 \% \) interest per year on the \( 40,000 \). So, the total interest equation is: \[0.021x + 0.065(40,000 - x) = 0.054 \times 40,000\]

Step 4: Simplify and Solve the Equation

First, calculate the total target interest: \[0.054 \times 40,000 = 2,160\]. Next, distribute and combine like terms in the equation: \[0.021x + 0.065(40,000 - x) = 2,160\] which simplifies to \[0.021x + 2,600 - 0.065x = 2,160\]. Then, combining like terms results in: \[-0.044x + 2,600 = 2,160\]. Solving for \( x \): \[-0.044x = 2,160 - 2,600\] \[-0.044x = -440\] Dividing both sides by \( -0.044 \): \[x = 10,000\]

Step 5: Determine the Amount in Mutual Fund

Since \( x = 10,000 \), the amount invested in the CD is \( 10,000 \). Therefore, the amount invested in the mutual fund account is \( 40,000 - 10,000 = 30,000 \).

Key Concepts

Algebraic Equations

Algebraic equations are foundational in solving investment problems. These equations let us find unknown values by setting up relationships based on given information. In Stephanie's case, we use an algebraic equation to balance her investments in a way that achieves her desired interest rate. We define variables and construct an equation to reflect the total interest earned from multiple sources. This process involves setting up expressions for each part of the investment and then solving the resulting equation to find the specific amounts to invest in each type of account.

Interest Calculation

Interest calculation is crucial in financial mathematics. It helps us determine how much money will be generated from investments over time. For Stephanie's problem, we calculate interest from both a certificate of deposit (CD) and a mutual fund. Each has a different interest rate. The formula to calculate interest is simple: multiply the principal amount by the interest rate. Therefore, if Stephanie invests \( x \) dollars in the CD at an interest rate of \( 2.1\text{\textperthousand} \) per year, the interest earned is \( 0.021x \). For the mutual fund with \( 6.5\text{\textperthousand} \) interest, she invests \( 40,000 - x \), leading to interest of \( 0.065(40,000 - x) \). Summing these interests and setting them equal to her combined desired interest completes the interest calculation.

Financial Mathematics

Financial mathematics involves using mathematical methods to solve financial problems and make investment decisions. Key concepts include present value, future value, interest rates, and annuities. In Stephanie's case, we combine these concepts with algebraic equations to determine the optimal investment strategy. This requires understanding how different interest rates affect the total return and how to allocate investments to meet specific financial goals. By modeling the problem with equations, financial mathematics provides a structured and precise approach to achieving desired outcomes.

Variable Definition

Defining variables is the first step in solving many mathematical problems, including investment scenarios. Variables represent unknown quantities that we need to find. For Stephanie's investment problem, we define two main variables: \( x \) for the amount invested in the CD and \( 40,000 - x \) for the amount in the mutual fund. Defining these variables helps us set up equations and track how each part of the investment contributes to the total interest earned. Clear variable definitions are crucial for making complex problems manageable and ensuring solutions are correct.