Problem 80 - 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. Cookbook Royalties Latasha Williams earned \(\$ 48,000\) from royalties on her cookbook. She paid a \(28 \%\) income tax on these royalties. The balance was invested in two ways, some of it at \(3.25 \%\) interest and some at \(1.75 \% .\) The investments produced \(\$ 904.80\) interest the first year. Find the amount invested at each rate.

Step-by-Step Solution

Answer

Invest $21,500 at 3.25% and $11,500 at 1.75%.

Step 1: Calculate the Tax Amount

First, calculate the amount Latasha pays in taxes. Since she is taxed at 28% on $48,000, the tax amount is given by:\[ \text{Tax Amount} = 48000 \times 0.28 \]

Step 2: Calculate the Balance after Tax

Subtract the tax amount from the total royalties to find the balance available for investment:\[ \text{Balance} = 48000 - \text{Tax Amount} \]

Step 3: Set up the System of Equations

Let's denote by \( x \) the amount invested at 3.25% and \( y \) the amount invested at 1.75%. We have two main equations:1. \( x + y = \text{Balance} \)2. \( 0.0325x + 0.0175y = 904.80 \) (representing the total interest earned).

Step 4: Substitute and Solve for One Variable

Solve the first equation for \( y \):\[ y = \text{Balance} - x \]Substituting this into the second equation gives:\[ 0.0325x + 0.0175(\text{Balance} - x) = 904.80 \]

Step 5: Solve for \( x \)

Simplify and solve the equation from Step 4:\[ 0.0325x + 0.0175 \times \text{Balance} - 0.0175x = 904.80 \]Combine like terms:\[ (0.0325 - 0.0175)x = 904.80 - 0.0175 \times \text{Balance} \]Finally, solve for \( x \).

Step 6: Find \( y \) Using \( x \)

With \( x \) found, use the equation \( y = \text{Balance} - x \) to find the value of \( y \).

Step 7: Verify Total Interest

Plug \( x \) and \( y \) back into the interest equation to verify that the total interest matches $904.80.

Key Concepts

Investment Problems

Investment problems are common scenarios where you must determine the optimal financial decisions, often involving choices on how to allocate funds. In this type of problem, you're given certain variables, like interest rates and initial investments, and tasked to find unknown values. To successfully find a solution, you'll typically need to use equations that tie together the given data points. For example, deciding how much money to invest in different accounts with varying interest rates requires careful setup of equations to represent the situation.
To break it down:

• Define how much total money you have to invest.
• Assign variables to unknown amounts.
• Create equations based on interest rates and the total money invested.
• Solve the equations to find out how much money should be invested in each account.

Properly setting up these equations is crucial, as it allows you to solve for unknowns and ensure your investments yield the desired results.

Income Tax Calculation

Calculating income tax is an essential step when planning investments, as taxes directly impact how much money you have available to allocate. The procedure is quite simple: start with the total income, apply the tax rate, and subtract the tax amount from the income to get the balance left for investment. In our exercise, Latasha Williams pays a 28% income tax on her $48,000 cookbook royalties.
Breaking it into steps:

• Begin with total earnings (e.g., $48,000).
• Calculate the tax amount by multiplying the earnings by the tax rate (28% in this case).
• Subtract the tax from the total to find what's left for investment.

Understanding this calculation helps you know how much real money you'll be working with post-taxes, crucial for informed investment decisions.

System of Linear Equations

A system of linear equations is a set of equations with multiple variables that you solve together. This mathematical approach is essential in solving investment problems like the one Latasha faces. She needs to determine how much to invest at two different interest rates. This setup requires creating a system of equations.
Here's how it's done:

• Define each variable (e.g., let \(x\) be the amount invested at 3.25% and \(y\) the amount at 1.75%).
• Use the information given to create equations, such as the total invested amount and the total interest earned.
• For Latasha, one equation reflects the sum of investments: \(x + y = \text{Balance}\).
• A second equation comes from the interest: \(0.0325x + 0.0175y = 904.80\).
• Use methods like substitution or elimination to solve these equations, finding values for \(x\) and \(y\).

With practice, solving these systems becomes a straightforward process, unlocking solutions to complex investment questions.

Interest Calculation

Interest calculation is the cornerstone of investment problems. It involves determining how much money an investment will earn over a specified period. The basic formula used is the simple interest formula: \(I = PRT\), where \(I\) is interest, \(P\) the principal amount, \(R\) the rate, and \(T\) the time. This formula assumes a consistent interest rate over the investment period.
In problems like Latasha's, she calculates interest from two investments.
To understand:

• Use the known interest rates for each investment portion.
• Apply the simple interest formula to calculate expected returns.
• Ensure the total interest from all investments matches the expected total, as a way to verify the calculations.

Mastering this means understanding not only how to calculate interest but how these calculations affect overall investment decisions. It's part of what makes managing real-life finances so powerful once you grasp the theory behind these calculations.