Problem 77 - A Graphical Approach to College Algebra

Question

Investment problems such as those in Exercises \(75-80\) 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, Mary Lynn Ellis 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

Mary deposited $10,000 at 2.5% and $20,000 at 3%.

Step 1: Define Variables

Let's say Mary deposits \( x \) dollars at a \( 2.5\% \) interest rate. Since she deposits twice as much at a \( 3\% \) interest rate, she deposits \( 2x \) at this rate.

Step 2: Write the Interest Formula

For each deposit, we apply the simple interest formula \( I = PRT \). For the first deposit, \( I_1 = x \times 0.025 \times 1 \). For the second deposit, \( I_2 = 2x \times 0.03 \times 1 \).

Step 3: Set up the Equation

The total interest from both investments should equal \( \$850 \). Therefore, form the equation: \( 0.025x + 0.06x = 850 \).

Step 4: Solve for x

Combine like terms in the equation to get \( 0.085x = 850 \). Divide both sides of the equation by \( 0.085 \) to solve for \( x \): \( x = \frac{850}{0.085} \).

Step 5: Calculate x

Perform the division: \( x = 10000 \). This means Mary deposited \( \$10,000 \) at \( 2.5\% \).

Step 6: Calculate Additional Deposits

Since twice as much is deposited at \( 3\% \), \( 2x = 20000 \). Mary deposited \( \$20,000 \) at a \( 3\% \) interest rate.

Key Concepts

Investment Problems

Investment problems often involve determining the optimal allocation of funds to achieve certain financial goals over a specific period. These problems can be approached using the simple interest formula, which is a straightforward way to calculate interest earned or payable on a principal amount. In the case of Mary Lynn Ellis, we broke down her retirement planning into two deposits with different interest rates. This allowed us to establish relationships between the amounts invested in each scenario.

To solve investment problems effectively, follow these steps:

• Define the variables involved, such as the principal amounts and their associated interest rates.
• Set up equations using the simple interest formula for each investment.
• Combine the results to create a system of equations, if needed, to represent the total acquired interest.
• Solve the system of equations to determine the unknown variables.

Using this method, we can unravel various investment allocations, as demonstrated in Mary's example. Understanding how to balance different interest rates is crucial for maximizing financial returns.

Retirement Planning

Retirement planning is a crucial financial strategy to ensure a comfortable future post-employment. In Mary's scenario, she strategically invested her savings in two different interest accounts to optimize her annual income. Planning for retirement involves evaluating potential incomes from savings and investments over time, taking into account risks, inflation, and individual financial goals.

It's important to diversify investments to achieve the right balance between risk and return. By depositing money in accounts with different interest rates, such as 2.5% and 3% in Mary's case, one can potentially increase overall interest income. Here are some key concepts in retirement planning:

• Understand the potential income from various investment types and their interest rates.
• Calculate predicted future value of savings using interest calculations.
• Reassess financial goals and investment strategies as your circumstances change.
• Monitor investments regularly to ensure they are on track to meet retirement needs.

Effective retirement planning requires careful consideration of all these elements to ensure financial security in later life.

Interest Rate Calculations

Interest rate calculations are vital in determining how much interest income or debt payment grows over time. The simple interest formula is one of the most accessible and straightforward tools for this purpose, especially for short term financial planning. The formula is expressed as:

\[ I = P \times R \times T \]

Where:

• \( I \) is the interest earned or payable.
• \( P \) is the principal amount invested or borrowed.
• \( R \) is the rate of interest per period expressed as a decimal.
• \( T \) is the time the money is invested or borrowed.

In the example of Mary Lynn Ellis, we applied this formula to her two different investment rates to determine her annual interest income. By correctly manipulating the formula, one can calculate the necessary principal or decide the feasibility of investment projects.

Understanding how to perform these calculations allows investors to make informed decisions about where to allocate funds to achieve desired outcomes. Always ensure to convert percentages into decimals before inserting them into the formula, as errors in this step can lead to inaccurate results.