Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Investment problems such as those in Exercises \(61-66\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\). where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Retirement Planning In planning her retirement, a woman deposits some money at \(2.5 \%\) interest with twice as much deposited at \(3 \% .\) Find the amount deposited at each rate if the total annual interest income is \(\$ 850\).

Step-by-Step Solution

Answer

\(\$10,000\) at 2.5%, \(\$20,000\) at 3%.

Step 1: Define Variables

Let the amount deposited at 2.5% be \( x \), and the amount deposited at 3% be \( 2x \). The simple-interest formula is \( I = P \cdot R \cdot T \). In our case, \( T = 1 \).

Step 2: Write Interest Formulas

The interest earned on the amount deposited at 2.5% is \( I_1 = x \cdot 0.025 \cdot 1 = 0.025x \).The interest earned on the amount deposited at 3% is \( I_2 = 2x \cdot 0.03 \cdot 1 = 0.06x \).

Step 3: Setup Total Interest Equation

According to the problem, the total interest from both deposits is \( \$850 \). Hence, we write the equation \( 0.025x + 0.06x = 850 \).

Step 4: Combine and Solve for x

Combine like terms in the equation: \( 0.025x + 0.06x = 0.085x \).So the equation becomes \( 0.085x = 850 \).Divide both sides by 0.085 to solve for \( x \):\[ x = \frac{850}{0.085} \approx 10000 \].

Step 5: Find Amount at 3%

The amount deposited at 3% is \( 2x = 2 \times 10000 = 20000 \).

Key Concepts

Investment Strategies

Investing money wisely is crucial for building wealth over time. A well-thought-out investment strategy can help you maximize returns while managing risks. In the context of the exercise, the strategy involves splitting the investment into two parts, each earning a different interest rate. This is done to optimize the total interest earned depending on the amount of money you have and the rates available. Investment strategies often include:

• Risk Assessment: Understanding how much risk you can tolerate will help in choosing the right investment types, like stocks, bonds, or savings accounts.
• Setting Goals: Establish clear short-term and long-term financial goals to guide your investment decisions.
• Diversification: Spreading investments across various assets to reduce risk.
• Regular Review: Monitoring your investments to ensure they are performing as expected and making adjustments as needed.

In the example, we see a basic approach by using simple interest calculation to maximize returns with available funds.

Retirement Planning

When planning for retirement, it's important to consider how much you need to save to maintain your desired lifestyle once you stop working. The exercise demonstrates a step towards this goal by focusing on how savings can grow through interest. Some key factors in retirement planning include:

• Estimating Retirement Needs: Calculate how much money you'll need annually once retired and how long you expect to be retired.
• Determining Contributions: Decide how much to save regularly to reach your retirement goals, taking into account potential interest earnings.
• Considering Inflation: Make sure to account for the rising cost of living over time.
• Choosing the Right Accounts: Different accounts offer tax benefits, such as IRAs and 401(k)s, which can impact savings growth.

The exercise helps to understand the importance of leveraging interest rates to boost savings, vital for future financial security.

Interest Rate Calculation

Interest rate calculation is key to understanding how your money grows over time. Simple interest, a straightforward method of interest calculation, is used in the exercise. It offers a foundational approach to interest earnings, where interest is earned only on the original amount deposited. The formula used is:\[ I = P \cdot R \cdot T \]Where:

• I: Interest earned
• P: Principal (initial amount of money)
• R: Rate of interest (expressed as a decimal)
• T: Time (in years)

In the given problem, the interest for both deposits was calculated separately and then combined to find the total interest. This process helps you understand how each portion of the investment contributes to overall earnings, highlighting the impact of different interest rates.

Algebraic Equations

Solving algebraic equations is an essential skill in tackling investment problems. It allows us to find unknown values which are crucial for making financial decisions. In this exercise, we used algebra to find the amounts deposited at two distinct interest rates. Here is the process broken down:

• First, define variables to represent unknown amounts. In the problem, \( x \) represented the amount at 2.5% interest, and \( 2x \) was used for the 3% interest.
• Setup equations based on the problem statement, utilizing known values like total interest.
• Combine like terms to simplify the equation, making it easier to solve. In the example, \( 0.025x + 0.06x = 850 \) simplifies to \( 0.085x = 850 \).
• Solve the equation by isolating the variable. Divide to find \( x \), and then calculate further steps if necessary, like doubling \( x \) for the second amount.

Understanding algebraic equations helps you take complex financial problems and break them down into more manageable parts, leading to effective solutions.