Problem 51 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

You borrow \(P\) dollars to buy a car and agree to repay the loan over \(t\) years at a monthly interest rate of \(i\) (expressed as a decimal). Your monthly payment \(M\) is given by either formula below. $$ M=\frac{P i}{1-\left(\frac{1}{1+i}\right)^{12 t}} \quad \text { or } \quad M=\frac{P i(1+i)^{12 t}}{(1+i)^{12 t}-1} $$ a. Show that the formulas are equivalent by simplifying the first formula. b. Find your monthly payment when you borrow \(\$ 15,500\) at a monthly interest rate of \(0.5 \%\) and repay the loan over 4 years.

Step-by-Step Solution

Answer

The monthly payment for a loan of \$15,500 at a rate of 0.5% over 4 years would be approximately \$359.64 and both formulas are found to be equivalent through algebraic simplification.

Step 1: Simplify the First Formula

First off, the goal is to simplify \(M = \frac{P i}{1 - (\frac{1}{1+i})^{12 t}}\). To do that, notice that \(1 - (\frac{1}{1+i})^{12 t} = \frac{(1+i)^{12 t} - 1}{(1+i)^{12 t}}\), which allows us to rewrite as \(M = \frac{P i (1 + i)^{12 t}}{(1 + i)^{12 t} - 1}\).

Step 2: Use the Simplified Formula to Calculate the Monthly Payment

Now, proceed to substitute the values \(P = 15500\), \(i = 0.5 / 100 = 0.005\) and \(t = 4\) into the formula: \(M = \frac{15500 * 0.005 * (1 + 0.005)^{12 * 4}}{(1 + 0.005)^{12 * 4} - 1}\). Calculate to get the value of the monthly payment \(M\).

Step 3: Compute the Monthly Payment

Perform the above calculations and you will find that \(M\) is approximately \$359.64. Thus, the monthly payment for a loan of \$15,500 at a monthly interest rate of 0.5% over a period of 4 years would be around \$359.64.

Key Concepts

Monthly Payment Formula

The monthly payment formula is used to calculate the fixed payment amount required to repay a loan with equal monthly installments over a specified time period. This formula takes into account the principal amount borrowed, the interest rate, and the number of payment periods. There are two common forms of the monthly payment formula, both of which are equivalent and can be used interchangeably:

• The first form is: \( M = \frac{P i}{1-(\frac{1}{1+i})^{12t}} \)
• The second form is: \( M = \frac{P i(1+i)^{12t}}{(1+i)^{12t}-1} \)

The formulas help in computing the monthly payment \(M\), where \(P\) is the loan principal, \(i\) is the monthly interest rate, and \(t\) is the number of years of the loan. These formulas are derived from the principles of compound interest.

Monthly Interest Rate

The monthly interest rate is critical in calculating loan payments. It represents the cost of borrowing expressed as a fraction of the principal on a monthly basis. To derive the monthly interest rate from an annual interest rate, you divide the annual rate by twelve. For instance, a 6% annual interest rate would result in a monthly interest rate of 0.5%, expressed as 0.005 in decimal form. Understanding how to correctly apply the monthly interest rate in calculations is essential because even small errors can significantly affect the monthly payment. As seen in the formula, the monthly interest rate \(i\) is compounded over the total number of payment periods, which can create a significant cumulative difference over a longer loan term.

Loan Amortization

Loan amortization is the process of paying off a loan over time through regular payments. Each payment covers both interest and a portion of the principal balance. As the loan period progresses, the share of the payment applied to the principal increases, while the interest portion declines. This dynamic ensures that the loan is fully paid off by the end of its term. Amortization schedules, often represented in charts or tables, provide borrowers with an overview of each payment's breakdown and the remaining balance after each installment. Understanding loan amortization is important for managing financial commitments and planning for future expenses. It also helps in assessing the true cost of a loan over its lifespan.