Problem 36 - Intermediate Algebra

Question

Entrepreneurs. Last year, a women's professional organization made two small- business loans totaling \(\$ 28,000\) to young women beginning their own businesses. The money was lent at \(7 \%\) and \(10 \%\) simple interest rates. If the annual income the organization received from these loans was \(\$ 2,560,\) what was each loan amount?

Step-by-Step Solution

Answer

The first loan was \$8000 at 7%, and the second was \$20000 at 10%.

Step 1: Define Variables

Let's assume that the first loan at 7% interest is \( x \) dollars. Then, the second loan at 10% interest is \( 28000 - x \) dollars.

Step 2: Set Up Interest Equations

The interest from the first loan is given by \( 0.07x \), and the interest from the second loan is \( 0.10(28000 - x) \). The total interest received is \$2560, so we can write the equation: \( 0.07x + 0.10(28000 - x) = 2560 \).

Step 3: Simplify the Equation

Let's distribute and simplify the equation: \[ 0.07x + 2800 - 0.10x = 2560 \]Combine like terms to get:\[ -0.03x + 2800 = 2560 \]

Step 4: Solve for x

Subtract 2800 from both sides to isolate the term with \( x \): \[ -0.03x = 2560 - 2800 \]\[ -0.03x = -240 \]Divide both sides by \(-0.03\):\[ x = \frac{-240}{-0.03} \]\[ x = 8000 \]

Step 5: Find the Second Loan

Since \( x \) is \\(8000, the first loan amount is \\)8000. The second loan is \( 28000 - 8000 = 20000 \). Thus, the second loan amount is \$20000.

Key Concepts

Interest Rates

Interest rates are crucial in finance, especially when it comes to loans. They represent the cost of borrowing money or the reward for saving money. In simple terms, an interest rate is a percentage charged on the total loan amount. For this exercise, the interest rates of 7% and 10% determine how much extra is paid or earned on the loans over the course of a year.

When you're dealing with simple interest, the interest earned or paid is calculated only on the principal amount—the initial amount of the loan or deposit. The formula for simple interest is:

• \[ I = P imes r imes t \] where:
  • \( I \) is the interest,
  • \( P \) is the principal amount,
  • \( r \) is the rate of interest per year (in decimal form),
  • \( t \) is the time period in years.

In this exercise, understanding how different interest rates affect the total interest earned can help solve the loan amounts quickly.

Remember, higher interest rates mean more cost when you borrow, but more earnings when you lend or invest.

Linear Equations

Linear equations can describe a myriad of real-world situations, such as balancing finances or predicting trends. In this context, linear equations help form relationships between variables—in this case, the different loan amounts and the interest rates.

This problem illustrates how linear equations can be used to form a balance: the interest from two different loans adds up to a known amount. The equation given is:

• \[ 0.07x + 0.10(28000 - x) = 2560 \]

This equation is set up by assigning the variable \( x \) to one of the unknown amounts and forming the expressions for interest based on the defined interest rates.

By solving linear equations like this one, you can find one of the unknown values (here, the amounts loaned at different rates), which is an essential skill in financial problem-solving.

Loan Problems

Loan problems often require organizing information and forming equations to find unknown values, such as individual loan amounts or interest earned. These problems not only test mathematical skills but also provide practical understanding of everyday financial transactions.

The main idea in such problems is balancing. Here, you're tasked with finding how much money was loaned at different rates. The systematic approach for solving loan problems follows these steps:

• Define the variables: Clearly specify what each variable represents.
• Set up equations: Use the given information to build equations. Here, calculate the interest using different rates and sum them to match the total interest.
• Solve the equations: Using algebraic techniques like combining like terms or isolating the variable, solve for unknowns.
• Verify the results: Always check that your answers satisfy the original conditions of the problem.

Understanding the essence of loan problems helps you not only in solving exercises but also in managing real-life financial decisions effectively.