Problem 61 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The small corporation borrowed \$216,000 at an 8% interest rate, \$505,000 at a 9% interest rate, and \$54,000 at a 10% interest rate.

Step 1: Define the variables

Let's label the amounts borrowed at each rate as follows: w is the amount borrowed at \(8\%\), x at \(9\%\), and y at \(10\%\). Let's express these definitions in dollars to match the units used in the exercise.

Step 2: Construct the equations

Now, we set up the equations based on the information given. The first equation represents the total amount borrowed: \(w + x + y = 775000\). The second equation represents the total interest owed at the end of the year: \(0.08w + 0.09x + 0.10y = 67500\). The third equation based on further specific information provided that amount borrowed at \(8\%\) is four times the amount borrowed at \(10\%\): \(w = 4y\). Now we have a system of three equations.

Step 3: Solve the system of equations

First, replacing the third equation into first gives us: \(4y + x + y = 775000\), after simplifying we get: \(x + 5y = 775000\). Replacing the third equation into second one gives us: \(0.08*4y + 0.09x + 0.10y = 67500\), after simplifying and converting to whole numbers we get: \(17x + 50y = 1500000\). Now solve these two equations for x and y using substitution or elimination method. From these calculations, we get \(x = 505000\) and \(y = 54000\). The value for w can be determined by replacing y into the third equation, which gives us \(w = 216000\).

Step 4: Verification

Plug these into the original equations to verify: Checking with first equation: \(216000 + 505000 + 54000 = 775000\), which is true. Then for the second equation, \(0.08*216000 + 0.09*505000 + 0.10*54000 = 67500\), which is true again. Hence, the solutions are verified.

Key Concepts

Overview of Systems of Equations in Finance

Systems of equations are a powerful tool used in finance mathematics to solve problems involving multiple variables.
They allow us to find unknown values by setting up several equations based on the given information. In this exercise, we deal with a financial scenario involving borrowed money at different interest rates.
To solve this problem, we establish a system of equations.

• The first equation represents the total amount of money borrowed across different rates.
• The second equation represents the total interest owed.
• The third equation accounts for specific conditions, like the relationship between the amounts borrowed at different rates.

Solving this system helps determine the exact amount borrowed at each interest rate, making these mathematical systems essential in finance.

Understanding Interest Calculation

Interest calculations are a key aspect of finance mathematics, allowing us to determine the additional money owed due to borrowing. In this example, we're dealing with simple interest, calculated using the formula:\[Interest = Principal \times Rate \times Time\]In our case, the total interest for each amount borrowed is represented like this:

• For the amount at 8%: \(0.08w\)
• For 9%: \(0.09x\)
• And 10%: \(0.10y\)

The problem specifies an annual interest, so the time is 1 year, simplifying our equations. The equation \(0.08w + 0.09x + 0.10y = 67500\) represents the sum of interests from all parts equaling the given total interest.
This method shows why it is crucial to understand interest rates and their impact when borrowing or investing.

Significance of Variable Definition

Defining variables is a fundamental step in solving systems of equations. This process helps to translate real-world situations into mathematical language everyone can understand.
In this exercise, each variable represents the amount borrowed at a specific rate:

• \(w\) for 8%
• \(x\) for 9%
• \(y\) for 10%

This careful definition keeps our system of equations organized and consistent.
It also ensures we can plug the values into our equations without confusion. When working with any problem, always define your variables first. It's like setting the stage before performing a play—it gets everything in order for the main action.

Verification of Solutions

Verifying solutions in finance mathematics is essential to ensure the calculations made are correct.
Once the values of the variables are found, they should be substituted back into the original equations to verify their correctness.
For instance, our calculated values in this problem are:

• \(w = 216000\)
• \(x = 505000\)
• \(y = 54000\)

To check their validity:- Add them together as per the first equation: \(216000 + 505000 + 54000 = 775000\).- Also, compute the total interest from these figures: \(0.08 \times 216000 + 0.09 \times 505000 + 0.10 \times 54000 = 67500\).Both checks confirm that the solutions satisfy the original conditions of the problem. This logical step ensures accuracy in finance-related calculations, preventing costly errors.