Problem 53 - College Algebra and Calculus: An Applied Approach

Question

Investment A real estate company borrows $$\$ 1,500,000 .$$ Some of the money is borrowed at \(7 \%\), some at \(8 \%\), and some at \(10 \%\) simple annual interest. How much is borrowed at each rate when the total annual interest is $$\$ 117,000$$ and the amount borrowed at \(8 \%\) is the same as the amount borrowed at \(10 \%\) ?

Step-by-Step Solution

Answer

The amount of money borrowed at each rate is: \(7 \%\) - \$300000, \(8 \%\) - \$600000, \(10 \%\) - \$600000.

Step 1: Set Up Variables

Let the amount of money borrowed at \(7 \%\) be \(x\), at \(8 \%\) be \(y\), and at \(10 \%\) be \(z\). Then, the total borrowed amount is represented by the equation: \(x + y + z = 1500000\).

Step 2: Construct the Equations

The annual interest is \$117000 which can be represented by the equation \(0.07x + 0.08y + 0.1z = 117000\). And the condition that the same amount is borrowed at \(8\%\) and \(10\%\) rates is represented with \(y = z\).

Step 3: Substitute One of the Variables

Substitute one of the equations that says 'y = z' into the other two. So now we have two equations with two variables, that are: \(x + 2y = 1500000\) and \(0.07x + 0.18y = 117000\).

Step 4: Solve the System

Solving these two equations, gives a solution \(x = 300000\), \(y = 600000\). Based on the given condition that the same amount is borrowed at \(8 \%\) and \(10 \%\) rates, it also means that \(z = 600000\).

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They consist of variables, coefficients, and constants, connected by arithmetic operations. These equations are pivotal in solving various problems requiring finding unknown values. For instance, in our investment problem, we express how much money is borrowed at different interest rates as an algebraic equation. By introducing variables such as x for the amount borrowed at 7%, y for 8%, and z for 10%, we transform the real-world scenario into an algebraic representation. This way, we can apply mathematical operations to solve for the values of x, y, and z, thus determining the amounts borrowed at each interest rate.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. Solving a system of equations means finding the values for the variables that satisfy all equations simultaneously. In our real estate company example, we are working with a system of three equations: the total amount borrowed, the total annual interest, and the condition that the same amount is borrowed at two different rates. To solve it, we use substitution or elimination strategies to reduce the system to a simpler form. In the step-by-step solution, y = z is used to substitute z with y, converting our three-variable system into a two-variable system. This is a basic method for solving systems, which simplifies the problem and guides us towards finding a solution more efficiently.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario. This process allows us to use the structure and language of mathematics to analyze and solve problems. In the exercise we’re examining, we're essentially using percentages representing interest rates, and total annual interest to develop a model that reflects the financial situation of the company. Once we have our mathematical model in place, which in this case includes variables and their relationships through algebraic equations, we can use it to perform calculations that yield valuable insights—like the specific amounts borrowed—at each interest rate. This is a powerful example of how mathematical modeling translates a practical financial question into a form that can be manipulated and solved using mathematical techniques.

Percentage Calculations

Percentage calculations are used to determine what proportion one quantity is of another one. They often appear in financial contexts, like the one in our example, where the interest rates are given in percent. It's essential to convert these percentages to decimal form before using them in algebraic equations. For instance, to find out the total annual interest from each rate, we multiply the principal amounts by their respective rates expressed as decimals (7% becomes 0.07, 8% becomes 0.08, and 10% becomes 0.10). By applying percentage calculations, we are able to express the total annual interest equation as 0.07x + 0.08y + 0.1z = 117000, where the coefficients represent the part of the interest that each amount contributes to the total. Understanding percentages is crucial because it's a common way to communicate ratios and proportions in a wide variety of contexts.