Problem 18 - College Algebra with Corequisite Support

Question

For exercises 17 and 18, use this scenario: A retired woman has $$\$ 50,000$$ to invest but needs to make $$\$ 6,000$$ a year from the interest to meet certain living expenses. One bond investment pays \(15 \%\) annual interest. The rest of it she wants to put in a CD that pays \(7 \%\). Set up and solve the equation for how much the woman should invest in each option to sustain a $$\$ 6,000$$ annual return.

Step-by-Step Solution

Answer

Invest \$31,250 in the bond and \$18,750 in the CD.

Step 1: Define Variables

Let the amount invested in the bond be \( x \). Then the amount invested in the CD is \( 50000 - x \), because the total investment is \$50,000.

Step 2: Write Expression for Total Interest

The interest from the bond investment is \( 0.15x \). The interest from the CD investment is \( 0.07(50000 - x) \). The total annual interest is given by the sum of these two amounts: \( 0.15x + 0.07(50000-x) \).

Step 3: Set Up Equation for Desired Return

The woman wants a total return of \$6,000 annually. Set up the equation: \( 0.15x + 0.07(50000-x) = 6000 \).

Step 4: Simplify the Equation

Distribute \( 0.07 \) in the second term: \( 0.15x + 3500 - 0.07x = 6000 \). Then combine like terms: \( 0.08x + 3500 = 6000 \).

Step 5: Solve for x

Isolate \( x \) by subtracting 3500 from both sides to get \( 0.08x = 2500 \). Solve for \( x \) by dividing both sides by 0.08: \( x = \frac{2500}{0.08} = 31250 \).

Step 6: Calculate Remaining Investment

The amount invested in the CD is \( 50000 - 31250 = 18750 \).

Key Concepts

Investment Strategies

Investment strategies are key to achieving financial goals, and understanding them helps in optimal capital allocation. In this scenario, the retired woman needs to decide how best to distribute her $50,000 investment to achieve a desired annual return of $6,000.

Here, multiple strategies can be considered:

• Diversification: Distributing the investment across different instruments can minimize risk. In this case, bonds and CDs offer varying risk profiles and returns.
• Risk Tolerance: Bonds commonly have higher interest rates and are perceived as higher risk compared to CDs. The investor's comfort with risk will dictate the proportion invested in each.
• Return Optimization: By investing strategically, the woman balances high returns from bonds with the stability of CDs to reach her financial target.

Developing a bespoke strategy based on these principles ensures that the investor's needs and risk preferences are met.

Interest Calculations

Interest calculations are essential for understanding how investments grow over time. They allow investors to anticipate the returns on different financial products. In this problem, the retired woman uses investments with fixed interest rates, which simplifies the calculations.

The formula for calculating interest is straightforward: \( \text{Interest} = \text{Principal} \times \text{Interest Rate} \)

• Bonds: The interest from the bonds is calculated as \(0.15x\), where \(x\) is the principal amount in bonds.
• CDs: The interest from the CDs is \(0.07(50000-x)\), the remaining amount after investing in bonds.

Adding these interests together shows the power of interest calculations in determining returns and achieving financial goals.

System of Equations

In solving investment problems, the system of equations can be a powerful tool. It allows investors to mathematically allocate funds to meet objectives. In this exercise, the woman sets up an equation to find out how much to invest in bonds and CDs.

The equation representing this scenario is: \(0.15x + 0.07(50000-x) = 6000\).

• Understanding Variables: Let \(x\) be the amount invested in bonds. Then \(50000 - x\) is in CDs.
• Goal: The target return is represented by 6000 in the equation.

By simplifying and solving this equation step by step, it shows how the system of equations facilitates strategic financial decision-making.

Financial Mathematics

Financial mathematics is integral in devising strategies for investment portfolios. It involves using mathematical theories and methods to solve financial problems. This scenario demonstrates financial mathematics by using algebraic techniques to find solutions.

Mathematics enables investors to:

• Optimize Returns: Calculating interest rates and using equations ensures optimal investment returns.
• Analyze Risks: Mathematical tools provide insight into potential risks with different financial products like bonds and CDs.
• Simplify Complex Problems: Breaking down the problem into equations makes sophisticated financial decisions manageable and clear.

Leveraging such mathematical concepts helps retirees, like the woman in this scenario, meet their financial needs in retirement.