Problem 66 - Contemporary Precalculus

Question

Use a recursively defined sequence to find the answer, as in Example 9. Lisa Chow is buying a condo. She takes out an 80,000 dollars mortgage for 30 years at \(6 \%\) annual interest. Her monthly payment is 479.64 dollars. (a) How much does she owe after one year ( 12 payments)? (b) How much interest does Lisa pay during the first five years? [Hint: The interest paid is the difference between her total payments and the amount of the loan paid off after five years \((80,000 \text { dollars }-\text { remaining balance }) .]\) (c) After 15 years, Lisa sells the condo and pays off the remaining mortgage balance. How much does she pay?

Step-by-Step Solution

Answer

(a) After one year, Lisa owes 78,198.96 dollars. (b) During the first five years, Lisa pays 22,593.37 dollars in interest. (c) When Lisa sells the condo after 15 years, she pays off the remaining mortgage balance of 61,810.82 dollars.

Step 1: Understand the mortgage payment formula

The monthly mortgage payment formula is given by: M = P * (r(1 + r)^n) / ((1 + r)^n - 1) Where: M = monthly payment P = principal amount (the loan amount) r = the monthly interest rate (annual interest rate / 12) n = the total number of payments (loan term in years times 12) In Lisa's case: P = 80,000 dollars Annual interest rate = 6% r = (6% / 12) / 100 = 0.06 / 12 = 0.005 Loan term = 30 years n = 30 * 12 = 360 payments Her monthly payment is already given as 479.64 dollars.

Step 2: Calculate the outstanding balance after one year (12 payments)

To find the outstanding balance after one year, we can use the outstanding principal formula: O = P * ((1 + r)^n - (1 + r)^k) / ((1 + r)^n - 1) Where: O = outstanding balance k = the number of payments so far In this case, k = 12 payments (one year). O = 80000 * ((1 + 0.005)^360 - (1 + 0.005)^12) / ((1 + 0.005)^360 - 1) O ≈ 78,198.96 dollars So, Lisa owes 78,198.96 dollars after one year.

Step 3: Calculate the interest paid during the first five years

To find the interest paid during the first five years, we'll first calculate the remaining balance after five years (60 payments). In this case, k = 60 payments (five years). O = 80000 * ((1 + 0.005)^360 - (1 + 0.005)^60) / ((1 + 0.005)^360 - 1) O ≈ 73,814.97 dollars The remaining balance after five years is 73,814.97 dollars. Now we can calculate the interest paid by finding the difference between the total payments and the amount of loan paid off. Total payments for five years = monthly payment * 60 Total payments = 479.64 * 60 = 28,778.40 dollars Amount paid off = Initial loan amount - remaining balance Amount paid off = 80,000 - 73,814.97 = 6,185.03 dollars Interest paid = Total payments - Amount paid off Interest paid = 28,778.40 - 6,185.03 = 22,593.37 dollars So, Lisa pays 22,593.37 dollars in interest during the first five years.

Step 4: Calculate the remaining balance after 15 years

To find the remaining balance when Lisa sells the condo after 15 years, we'll find the remaining balance after 180 payments (15 years). In this case, k = 180 payments. O = 80000 * ((1 + 0.005)^360 - (1 + 0.005)^180) / ((1 + 0.005)^360 - 1) O ≈ 61,810.82 dollars So, the remaining balance after 15 years is 61,810.82 dollars.

Step 5: Conclusion

Now we have the answers to all three parts of the problem: (a) Lisa owes 78,198.96 dollars after one year (12 payments). (b) Lisa pays 22,593.37 dollars in interest during the first five years. (c) Lisa pays off 61,810.82 dollars remaining mortgage balance after 15 years when she sells the condo.

Key Concepts

Mortgage Payment Formula

Understanding the mortgage payment formula is essential for anyone looking to buy property using a loan. This formula helps determine the monthly payment amount, which is a critical factor in budgeting for a home purchase.

The formula is: \[ M = P \times \frac{r(1 + r)^n}{((1 + r)^n - 1)} \]

where \( M \) represents the monthly payment, \( P \) stands for the principal (loan amount), \( r \) is the monthly interest rate, and \( n \) signifies the total number of payments. By substituting the relevant values, borrowers can easily figure out their monthly mortgage obligation. It's the starting point in managing a long-term financial commitment.

Outstanding Balance Calculation

The outstanding balance calculation reveals how much of the principal still remains after making a series of payments. It's a glimpse into the future, showing the loan's progression and how much debt is left to pay.

The concept hinges on this core formula: \[ O = P \times \frac{((1 + r)^n - (1 + r)^k)}{((1 + r)^n - 1)} \]

where \( O \) is the outstanding balance, \( P \) is the principal, \( r \) is the monthly interest rate, \( n \) is the total number of payments, and \( k \) is the number of payments made so far. This calculation is crucial for planning, as it allows you to understand the loan's trajectory and adjust financial strategies accordingly.

Interest Payment Calculation

A critical aspect of managing a mortgage is understanding the interest payment calculation. Interest is the cost of borrowing money and constitutes a significant portion of the total amount repaid over the life of the loan.

To determine the interest paid over a period, the borrower subtracts the amount of the loan paid off (principal repayment) from the total payments made. This is expressed as:

\[ \text{Interest paid} = (\text{Total payments}) - (\text{Initial loan amount} - \text{Remaining balance}) \]

Calculating the interest paid allows borrowers to see how much they are spending on interest alone, which can often be a motivator to increase payments and shorten the loan term, saving money in the long term.