Problem 83 - College Algebra

Question

How much are your monthly payments on a loan? If \(P\) is the principal, or amount borrowed, \(i\) is the monthly interest rate, and \(n\) is the number of monthly payments, then the amount, \(A,\) of each monthly payment is $$A=\frac{P i}{1-\frac{1}{(1+i)^{n}}}$$ a. Simplify the complex rational expression for the amount of each payment. b. You purchase a 20,000 dollars automobile at \(1 \%\) monthly interest to be paid over 48 months. How much do you pay each month? Use the simplified rational expression from part (a) and a calculator. Round to the nearest dollar.

Step-by-Step Solution

Answer

The monthly payment is \(\$\)413.

Step 1: Simplify the complex rational expression

The given formula for the loan payment is a complex fraction. To simplify it, we start by multiplying both the numerator and denominator by \((1+i)^{n}\).\nThis leads to:\n\[A=\frac{P i (1+i)^{n}}{(1+i)^{n}-1}\]

Step 2: Substitution/Calculation

Now, plug the given values for \(P = 20,000\), \(i=0.01\), and \(n=48\) into the simplified formula and solve for \(A\). That gives us:\n\[A=\frac{20000 * 0.01 *(1+0.01)^{48}}{(1+0.01)^{48}-1}\]

Step 3: Rounding to the nearest dollar

After performing the calculation, you will obtain a decimal number. Round this number to the nearest dollar to get the monthly payment.

Key Concepts

Understanding Complex Rational Expressions

In mathematics, complex rational expressions like the one used in loan payment calculations can seem intimidating. But breaking them down makes them manageable. The expression \(A=\frac{P i}{1-\frac{1}{(1+i)^{n}}}\) is a blended fraction. The main idea is that you solve it by removing the fraction in the denominator.

You can do this by multiplying both the top and bottom by \((1+i)^{n}\). This simplifies the expression to \(A=\frac{P i (1+i)^{n}}{(1+i)^{n}-1}\).

• This approach helps you focus on simple operations like multiplication and subtraction.
• Removing nested fractions makes calculations easier.

Once simplified, it's much easier to plug in numbers and find solutions, keeping errors at bay.

Calculating the Monthly Interest Rate

The monthly interest rate \(i\) is a core component of the loan payment formula. It's crucial to know how to convert an annual interest rate to a monthly one, if necessary.

Simply divide the annual rate by 12. For example, a 12% annual rate becomes 1% per month or 0.01 in decimal form.

• Use the decimal form in calculations to avoid confusion.
• Always ensure the interest rate aligns with the payment period, like months.

Understanding the monthly rate properly ensures your calculations are accurate.

Principal Amount Explained

The principal amount, denoted as \(P\) in the formula, is the initial sum of money borrowed or invested. It's the foundation for calculating repayments.

In our sample problem, the principal is $20,000 for a car purchase.

• Ensure the principal amount is clearly defined before starting calculations.
• Remember that it affects interest calculations and total repayment.

Recognizing the principal's role can aid in understanding better how loans work.

Effective Calculator Usage for Loan Payments

Using a calculator efficiently can greatly simplify solving complex financial formulas. For loan payments, input each component carefully:

First, solve the exponent, like \((1+i)^{n}\). Then, tackle the fractional parts.

• Double-check each entry to ensure accuracy.
• Rounding should be the last step after fully solving the equation.

Use the memory functions if your calculator has them; they can save intermediate results to reduce error. By mastering calculator skills, you ensure calculations don't overwhelm you, and the results remain reliable.