Problem 31 - Introductory Algebra for College Students

Question

Involve dual investments. A bank loaned out 120,000 dollar part of it at the rate of \(8 \%\) annual mortgage interest and the rest at the rate of \(18 \%\) annual credit card interest. The interest received on both loans totaled 10,000 dollar How much was loaned at each rate? Organize your work in the following table. (TABLE CAN NOT COPY))

Step-by-Step Solution

Answer

The amount loaned at the rate of 8% annual mortgage was $116,000 and the amount loaned at 18% annual credit card rate was $4,000.

Step 1: Set Up the Equations

Let's denote the amount loaned at 8% annual mortgage interest as \(X\) and the amount loaned at 18% annual credit card interest as \(Y\). Therefore, the sum of \(X\) and \(Y\) should be $120,000 which gives us our first equation: \(X + Y = 120,000\) \n\nThe interest earned from these two loans is $10,000. The interest earned from each of these amounts can be expressed as 0.08*X for the first loan and 0.18*Y for the second loan. This gives us the second equation: \(0.08X + 0.18Y = 10,000\)

Step 2: Solve the Equations

We can solve this system by either the substitution or the elimination method. Here, we use the elimination method. Multiply the first equation by 0.08 which will give us: \(0.08X + 0.08Y = 9,600\). By subtracting this new equation from the second equation, we get \(0.1Y = 400\), which simplifies to \(Y = 4,000\). Substituting \(Y\) in the first equation, we get \(X = 120,000 - 4,000 = 116,000\)

Step 3: Verify the Solution

Substitute \(X = 116,000\) and \(Y = 4,000\) back into original equations to verify that they satisfy the conditions mentioned in the problem.

Key Concepts

Interest Calculation

Interest calculation is essential when dealing with loans, as it helps determine the amount of money owed over time. To calculate interest, especially simple interest, you generally use the formula: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] where:

• Principal is the initial amount of money lent or borrowed.
• Rate is the annual interest rate (expressed as a decimal).
• Time is the duration for which the money is borrowed or lent.

In our exercise, two different interest rates were used: 8% for a mortgage and 18% for a credit card. This means the calculation will involve determining how much of the total amount, $120,000, is associated with each rate, thereby determining the total interest accrued.

Elimination Method

The elimination method is a powerful technique for solving systems of linear equations by eliminating one of the variables. To use the elimination method, you need to combine two equations to cancel out one variable, allowing you to solve for the remaining variable. Here's how it typically works:

• Align your equations so they have similar terms.
• If necessary, multiply one or both equations to obtain identical coefficients for one of the variables.
• Subtract (or add) equations to eliminate one variable.
• Solve for the remaining variable.
• Substitute back into one of the original equations to find the other variable.

In this exercise, the method was used by first aligning the equations and scaling them as needed to eliminate the variable, in this case, solving for the balance of the loans at given interest rates.

Mortgage Interest

Mortgage interest refers to the interest charged on a loan used to purchase a property. This interest can vary widely in rates, terms, and conditions. Typically, mortgage interest is lower than unsecured borrowing options like credit cards, as it's secured against the property being purchased. In our example, the bank loaned out part of the funds at an 8% annual mortgage interest rate. Here's why mortgage interest tends to be lower:

• Collateral: As the loan is secured by real estate, it reduces risk for the lender.
• Loan Terms: Mortgages usually involve longer repayment terms, spreading out the interest payment.
• Market Rates: The rates reflect economic factors, such as national interest rates and inflation.

Understanding mortgage interest is crucial for estimating costs in home financing and recognizing how mortgage structure influences overall financial planning.

Credit Card Interest

Credit card interest is the fee charged by a credit card issuer when you carry a balance past the due date. It tends to be higher than other forms of borrowing, such as mortgages, due to the unsecured nature of credit card debt. Here’s why credit card interest rates are usually higher:

• No Collateral: Unlike a mortgage that is backed by real estate, credit cards are unsecured forms of debt.
• Short-Term Lending: Credit cards offer short-term balances with a revolving nature, reflecting higher volatility and risk for lenders.
• Risk & Cost: The costs and risks associated with managing a wide range of borrowers.

In the provided exercise, a part of the funds was loaned at a high 18% credit card interest rate, reflecting the higher risk and management cost associated with this form of credit. Understanding how credit card interest is applied can greatly influence the management and repayment strategy for personal finances.