Problem 37 - Intermediate Algebra

Question

Paula used some of the money that she received from an inheritance to invest in a certificate of deposit paying \(7 \%\) annual interest and the rest of the money in a promising biotech company offering an annual return of \(10 \%\). She invested twice as much in the \(10 \%\) investment as she did in the \(7 \%\) investment. Her combined annual income from the two investments was \(\$ 4,050 .\) A. How much did she invest in each account? B. How much did she inherit?

Step-by-Step Solution

Answer

Paula invested \(\$15,000\) in the \(7\%\) account and \(\$30,000\) in the \(10\%\) account, inheriting \(\$45,000\).

Step 1: Define Variables

Let's define the variables: Let \( x \) be the amount Paula invested in the certificate of deposit at \(7\%\). Since she invested twice that amount in the biotech company, the investment in the company at \(10\%\) is \( 2x \).

Step 2: Set Up the Equation for Total Income

The annual income from the \(7\%\) investment is \(0.07x\), and from the \(10\%\) investment is \(0.10 \times 2x = 0.20x\). The combined annual income from both investments is \( \$4050 \). So our equation is: \[ 0.07x + 0.20x = 4050 \]

Step 3: Solve the Equation

Combine the terms in the equation: \[ 0.27x = 4050 \]Solve for \( x \) by dividing both sides by \( 0.27 \): \[ x = \frac{4050}{0.27} \approx 15000 \]Thus, Paula invested \( \$15,000 \) in the \(7\%\) certificate of deposit.

Step 4: Calculate the Investment in Biotech Company

Since she invested twice as much in the biotech company, the investment there is \( 2x = 2 \times 15000 = 30000 \). So, she invested \( \$30,000 \) in the \(10\%\) biotech investment.

Step 5: Calculate Total Inherited Amount

The total inherited amount is the sum of both investments: \[ 15000 + 30000 = 45000 \]Thus, she inherited \( \$45,000 \).

Key Concepts

Simple Interest

When dealing with investments like those in Paula's exercise, simple interest is a key concept to understand. Simple interest is calculated on the principal amount, which is the initial amount you invest, rather than on the accumulated interest over time. This makes it straightforward to understand and easy to calculate.
To find the interest earned from a single investment, you use the formula: \[ I = P imes r imes t \] where:

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

In Paula's case, since she invested for one year, the time \( t \) is \( 1 \). Therefore, the calculations are straightforward, only involving the multiplication of principal and interest rate. Each part of her investment, the 7% and the 10%, generates some income that together sums up to her total annual gain.

Algebraic Equations

Algebraic equations are a powerful tool to solve investment word problems by representing relationships mathematically.

In the exercise, we define \( x \) as the amount invested at 7%. Because Paula invested twice this amount at 10%, we can set the investment in the biotech company as \( 2x \). The total interest earned from both investments is the sum of each interest portion: which can be represented by the equation \((0.07x + 0.20x = 4050)\).

The reason these algebraic equations are so effective is that they allow you to combine and solve parameters to understand how much was invested in each account. It logically brings together different elements of the problem, leading to a conclusion about the initial investment amounts Paula made.

Investment Strategies

In the world of finance, choosing the right investment strategy is crucial for maximizing returns while considering risk.

Paula's approach in the problem showcases a classic strategy: diversification. By splitting her inheritance into two different investments, she mitigates risk by not "putting all her eggs in one basket."

• The certificate of deposit offers a stable, secure return, albeit at a lower percentage.
• The investment in the biotech company, though offering a higher return at 10%, likely comes with greater risk.

Diversification as an investment strategy helps investors stabilize their returns. The secure investment provides a steady income, while the riskier one offers a chance to earn more. Paula's strategy balanced risk with opportunity in a strategic manner for optimum financial health.