Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

A small corporation borrowed \(\$ 775,000\) to expand its software line. Some of the money was borrowed at \(8 \%,\) some at \(9 \%,\) and some at \(10 \% .\) How much was borrowed at each rate, given that the annual interest was \(\$ 67,500\) and the amount borrowed at \(8 \%\) was four times the amount borrowed at \(10 \% ?\)

Step-by-Step Solution

Answer

The corporation borrowed \$500000 at 8%, \$225000 at 9% and \$50000 at 10%.

Step 1: Define Variables

Let's denote the amount borrowed at 8% as \( x \), at 9% as \( y \), and at 10% as \( z \).

Step 2: Read the problem and form equations

From the problem we know that the total amount borrowed is \$775,000. So our first equation will be \( x + y + z = 775,000 \). The problem also mentions that the annual interest was \$67,500. Interpret this as 8% of the amount borrowed at 8%, plus 9% of amount borrowed at 9% and 10% of amount borrowed at 10% i.e. \( 0.08x + 0.09y + 0.10z = 67,500 \). From the third condition, we get that the amount borrowed at 8% is four times the amount borrowed at 10% or \( x = 4z \).

Step 3: Solve the system of linear equations

Now we have a system of three equations: \( x+y+z = 775,000 \), \( 0.08x + 0.09y + 0.10z = 67,500 \) and \( x = 4z \). Substituting \( x = 4z \) into the other two equations, we can solve for \( y \) and \( z \), and then obtain \( x \).

Step 4: Calculate final amounts

By doing the above calculations, you will find that \( x = \$500000 \), \( y = \$225000 \), and \( z = \$50000 \). Therefore, the corporation borrowed \$500000 at 8%, \$225000 at 9% and \$50000 at 10%.

Key Concepts

Linear Equations

Linear equations form the backbone of solving algebraic problems where variables represent unknown values. In this exercise, the variables are used to signify the amounts borrowed at different interest rates. A linear equation is an equation involving one or more variables where the power of the variable(s) is one.

The given problem entails forming a set of linear equations based on conditions provided:

• Total Borrowed Amount: The first equation is based on the total sum borrowed being equal to \(775,000. This is expressed as \( x + y + z = 775,000 \), where \( x, y, \) and \( z \) represent amounts borrowed at 8%, 9%, and 10% respectively.
• Interest Calculation: Interest from each loan adds up to \)67,500. This gives another equation: \( 0.08x + 0.09y + 0.10z = 67,500 \).
• Ratio of Amounts Borrowed: The amount borrowed at 8% is four times the amount borrowed at 10%, leading to \( x = 4z \).

Using these equations, a system of linear equations is formed, and solving it allows us to find the unknown amounts.

Interest Calculation

Interest calculation is an essential part of financial mathematics, helping to determine how much cost or profit is generated by borrowing or investing money. In this problem, we need to understand how each part of the borrowed amount contributes to the total interest.

We calculate interest by multiplying the borrowed amount by the interest rate. For example, if the amount borrowed at 8% is \( x \), the interest generated from this amount will be \( 0.08x \). Here are the contributions to the total annual interest:

• The contribution from 8% interest is \( 0.08x \).
• The contribution from 9% interest is \( 0.09y \).
• The contribution from 10% interest is \( 0.10z \).

The sum of these individual interests should match the given total interest, which is $67,500. By solving the equation \( 0.08x + 0.09y + 0.10z = 67,500 \), we can help determine how much was borrowed at each rate.

Algebraic Problem Solving

Algebraic problem solving involves using algebraic techniques and processes to find unknown values from given equations. This exercise showcases the use of such techniques.Here’s the systematic approach to solving the problem:

• First, identify variables for unknowns with clear relations (e.g., \( x = 4z \)).
• Write down all possible equations that reflect the problem's conditions.
• Next, solve the system of equations. This can be done by substitution or elimination methods.
• For our problem, substituting \( x = 4z \) into the other equations simplifies the equations:
• Solve the resulting equations for \( y \) and \( z \).
• Once \( z \) is found, use \( x = 4z \) to find \( x \).
• Verify the solutions to ensure they satisfy all original conditions.

By following these steps, the unknown amounts become known, offering clarity and reliability in problems involving algebraic solutions.