Problem 15 - Precalculus Mathematics for Calculus

Question

A woman wants to borrow \(\$ 12,000\) to buy a car. She wants to repay the loan by monthly installments for 4 years. If the interest rate on this loan is \(10 \frac{1}{2} \%\) per year, compounded monthly, what is the amount of each payment?

Step-by-Step Solution

Answer

The monthly payment is approximately \(\$307.88\).

Step 1: Convert Interest Rate to Monthly Rate

The annual interest rate is given as \(10 \frac{1}{2}\%\) per year. First, convert this annual rate to a monthly rate by dividing it by 12. The annual rate as a decimal is \(10.5\% = 0.105\). Thus, the monthly interest rate \(r\) is \( \frac{0.105}{12} \approx 0.00875\).

Step 2: Determine Total Number of Payments

The loan term is 4 years, and the woman will make monthly payments. Calculate the total number of monthly payments by multiplying the number of years by 12. Therefore, the total number of payments \(n\) is \(4 \times 12 = 48\).

Step 3: Use Annuity Formula for Payment Calculation

The formula for the monthly payment \(M\) on an amortizing loan is given by: \[M = \frac{P \times r}{1 - (1 + r)^{-n}}\]where \(P = \$12,000\) is the principal amount, \(r = 0.00875\) is the monthly interest rate, and \(n = 48\) is the total number of payments.

Step 4: Substitute the Values into the Formula

Plug the values into the annuity formula: \[M = \frac{12000 \times 0.00875}{1 - (1 + 0.00875)^{-48}}\]

Step 5: Calculate Monthly Payment

First, calculate the denominator part:\[(1 + 0.00875)^{-48} \approx 0.65704\]Then, the whole denominator:\[1 - 0.65704 = 0.34296\]Finally, substitute these into the formula:\[M = \frac{12000 \times 0.00875}{0.34296} \approx 307.88\]

Key Concepts

Interest Rate Conversion

When dealing with loans, it's common to encounter interest rates provided on a yearly basis; these are often referred to as the annual percentage rate (APR). However, for calculations that involve monthly repayments, it's crucial to convert these annual rates into a monthly rate. This process is known as interest rate conversion.
To convert an annual interest rate to a monthly one, you simply divide the annual rate by the number of months in a year, which is 12. In our example, the interest rate is given as \(10.5\%\) per annum. As a decimal, this is \(0.105\). Thus, when we divide by 12, we find the monthly interest rate:

• Annual rate: \(0.105\)
• Monthly rate: \(\frac{0.105}{12} \approx 0.00875\)

After converting to a monthly rate, this allows us to apply it directly to each monthly payment calculation. This is essential to ensure accuracy over the entire repayment period.

Annuity Formula

The annuity formula is a key tool used in calculating regular payments on a loan or investment. For loans, especially those amortized over time like car loans or mortgages, this formula helps determine the fixed periodic payment amount.
The formula can be expressed as follows:\[M = \frac{P \times r}{1 - (1 + r)^{-n}}\]Here, \(M\) represents the monthly payment, \(P\) is the principal amount (the initial amount borrowed), \(r\) is the periodic interest rate (in this context, the monthly interest rate), and \(n\) is the total number of payments. By using this formula, we can ascertain the precise amount that must be paid each month to service the debt completely by the end of the loan term. This allows the borrower to plan their finances effectively, as they know exactly how much they will need to pay each month.
Using the calculated values from our example, we have:

• Principal, \(P = \$12,000\)
• Monthly interest rate, \(r = 0.00875\)
• Total payments, \(n = 48\)

These values are then plugged into the formula to determine the monthly payment required.

Monthly Payment Calculation

Calculating the monthly payment for a loan using the annuity formula requires several steps. This is often done by calculators for speed, but understanding the process will give a deeper insight into how loans work.
Firstly, the values are substituted into the annuity formula:\[M = \frac{12000 \times 0.00875}{1 - (1 + 0.00875)^{-48}}\]The formula's complexity lies in the calculation of the denominator. Let's break it down:

• Calculate \((1 + r)^{-n}\):
  The term \((1 + 0.00875)^{-48}\) results in approximately \(0.65704\).
• Subtract from 1:
  The expression becomes \(1 - 0.65704 = 0.34296\).

Then, with a fully solved denominator:
Finally, solve the entire formula:\[M = \frac{12000 \times 0.00875}{0.34296} \approx 307.88\]This calculation yields the monthly payment of approximately \(\$307.88\). This amount is what needs to be paid every month for the loan's duration, which in this case, is 4 years.