Problem 61 - 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. Real-Estate Financing A man wishes to sell a piece of property for \(\$ 240,000 .\) He wants the money to be paid off in two ways: a short-term note at \(6 \%\) interest and a longterm note at \(5 \%\). Find the amount of each note if the total annual interest paid is \(\$ 13,000\)

Step-by-Step Solution

Answer

Short-term note: \(\$100,000\); Long-term note: \(\$140,000\).

Step 1: Define Variables

Let's define the variables for this problem. Let \( x \) be the amount of money invested in the short-term note at 6% interest, and \( y \) be the amount of money invested in the long-term note at 5% interest. So, we have two equations: \( x + y = 240,000 \) (total property price) and \( 0.06x + 0.05y = 13,000 \) (total interest).

Step 2: Set Up The Equations

We establish the relationships between the finances using the equations from Step 1:1. \( x + y = 240,000 \) (Equation 1).2. \( 0.06x + 0.05y = 13,000 \) (Equation 2).These represent the total investment amount and the total interest respectively.

Step 3: Solve for y in Equation 1

Using Equation 1, we express \( y \) in terms of \( x \):\[ y = 240,000 - x \]

Step 4: Substitute y in Equation 2

Substitute the expression for \( y \) from Step 3 into Equation 2:\[ 0.06x + 0.05(240,000 - x) = 13,000 \]

Step 5: Simplify and Solve for x

Expand and simplify the equation from Step 4:\[ 0.06x + 12,000 - 0.05x = 13,000 \]Combine like terms:\[ 0.01x + 12,000 = 13,000 \]Subtract 12,000 from both sides:\[ 0.01x = 1,000 \]Divide both sides by 0.01:\[ x = 100,000 \]

Step 6: Solve for y

Now that we have the value for \( x \), substitute \( x = 100,000 \) back into the expression for \( y \):\[ y = 240,000 - x = 240,000 - 100,000 = 140,000 \]

Step 7: Conclusion

The short-term note is \( \\(100,000 \) and the long-term note is \( \\)140,000 \).

Key Concepts

Linear Equations

Linear equations are the foundation for solving many mathematical problems, including financial ones. They involve variables, constants, and an equal sign, forming a straight line when graphed. The solutions to these equations help us find values of unknowns that make the equations true.

In the context of the exercise, two linear equations were used to determine the amounts invested in short and long-term notes. The first equation came from the total property price, while the second was based on the total annual interest. These are known as simultaneous equations because they are solved together. The process typically involves:

• Defining the variables
• Setting up equations based on given information
• Substituting one equation into the other to find solutions

Linear equations are highly versatile, allowing for clear and systematic solutions to diverse problems.

Investment Problems

Investment problems are a type of financial problem that require determining how money should be allocated to achieve certain financial goals. These often involve calculating the returns from different investments over a specific period of time.

In the exercise, the man wanted to divide his investment, the sale of a property, into two parts to maximize returns. Such problems often involve:

• Determining the total investment amount
• Setting financial targets, like desired interest return
• Choosing different interest rates for various investment types

Solving investment problems can help in making informed decisions about financial strategies. Whether considering short-term or long-term gains, understanding the fundamental components involved is crucial.

Financial Mathematics

Financial mathematics focuses on modeling financial markets and assessing the value of investments over time. It uses various mathematical tools and principles to solve practical financial problems.

Simple interest is a basic concept in financial mathematics, defined by the formula:\[ I = PRT \]where:

• \( I \) is the interest earned
• \( P \) is the principal amount or initial investment
• \( R \) is the annual interest rate (expressed as a decimal)
• \( T \) is the time in years

This formula is used to calculate the interest earned on an investment based on specific parameters. In our exercise, it was modified to derive the total interest for investments in short-term and long-term notes. Understanding simple interest and applying it correctly is fundamental for making sound financial decisions. Mastery of financial mathematics ensures accurate calculations and feasible investment plans.