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Chapter 8: Problem 7

Linda wants to purchase a Leisure Heights condominium apartment. She will borrow \(\$ 100,000\) from the Duchess Savings Bank. The bank is presently offering a \(30-\) year fixed rate mortgage with an APR of 7.1\(\% .\) Her monthly maintenance fee will be \(\$ 310 .\) a. What is the monthly mortgage payment to the nearest cent? b. What will be her combined monthly payment?

Short Answer

Expert verified
a. The monthly mortgage payment to the nearest cent is approximately $668.43. b. The combined monthly payment is $978.43.

Step by step solution

01

Calculate the monthly interest rate

To get the monthly interest rate, divide the APR by 12 since there are 12 months in a year: \n\n\(i = \frac{APR}{12} = \frac{7.1}{12} = 0.00592\), approximately.
02

Determine the number of payments

Calculate the number of payments by multiplying the number of years by 12 (since there are 12 months in a year): \n\n\( n = years \times 12 = 30 \times 12 = 360 \) payments.
03

Calculate the monthly mortgage payment

Use the formula to find out the monthly payment: \n\n\(P = \frac{PV \times i \times (1 + i)^n}{(1 + i)^n - 1}\), \n\nwhere: \n- P is the monthly payment, \n- PV is the present value or the amount of loan, which is $100,000, \n- i is the monthly interest rate, \n- n is the total number of payments. \n\nPlug in the values to the formula to get: \n\n\(P = \frac{100000 \times 0.00592 \times (1 + 0.00592)^{360}}{(1+0.00592)^{360} - 1} = $668.43\), approximately.
04

Calculate combined monthly payment

The combined monthly payment includes both the mortgage payment and the maintenance fee. Simply add the two amounts to get the combined payment: \n\nCombined Payment = Mortgage + Maintenance Fee = $668.43 + $310 = $978.43

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

APR (Annual Percentage Rate)
APR, or Annual Percentage Rate, expresses the cost of borrowing money on an annual basis. It includes interest rate along with other charges or fees involved in obtaining the loan.

It's essential for Linda, or any borrower, to understand APR as it gives a broader picture of the loan cost compared to the simple interest rate. APR is used to compare different loan offers, as a lower APR can lead to significant savings over the life of a loan.

When calculating a monthly mortgage payment, the APR is converted to a monthly interest rate since payments are typically made monthly. This is done by dividing the APR by 12, thereby obtaining the rate applicable for the monthly amortization schedule.
Fixed Rate Mortgage
A fixed rate mortgage, as its name suggests, has a set interest rate that doesn't change throughout the term of the loan. This means that Linda's payments for the principal and interest will remain the same each month for the life of the mortgage.

This consistency makes budgeting easier for homeowners, as they don鈥檛 have to worry about varying payment amounts that could result from interest rate fluctuations common to adjustable-rate mortgages. Here, Linda鈥檚 30-year mortgage will have the same monthly payment at the fixed APR of 7.1%, which offers her financial predictability over the long term.
Loan Amortization
Loan amortization is the process through which a loan is paid off over time in regular, equal payments. With each payment Linda makes, a portion will go towards the interest accrued and the remaining amount will be applied to the principal balance.

Initially, a larger part of the payment goes toward interest, but as the loan balance decreases, more of the payment goes toward paying down the principal. This method of loan repayment ensures that the loan is gradually paid in full by the end of the fixed period, which in Linda's case is 30 years.
Monthly Interest Rate
The monthly interest rate is essentially the APR converted to a monthly figure. Since interest rates are usually quoted on an annual basis and mortgage payments are made monthly, the conversion is necessary for calculating the regular payment amount.

To obtain Linda's monthly interest rate, the APR of 7.1% is divided by 12, equating to approximately 0.00592 or 0.592%. This monthly rate is relevant when determining the amount of interest that needs to be paid in each installment as part of the amortization of the loan.

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Most popular questions from this chapter

If you borrow \(120,000 at an APR of 7% for 25 years, you will pay \)848.13 per month. If you borrow the same amount at the same APR for 30 years, you will pay $798.36 per month. a. What is the total interest paid on the 25-year mortgage? b. What is the total interest paid on the 30-year mortgage? c. How much more interest is paid on the 30-year loan? Round to the nearest dollar. d. If you can afford the difference in monthly payments, you can take out the 25-year loan and save all the interest from part c.What is the difference between the monthly payments of the two different loans? Round to the nearest dollar.

Jim is taking out a $135,000 mortgage. His bank offers him an APR of 6.25%. He wants to compare monthly payments on a 20- and a 30-year loan. Find, to the nearest dollar, the difference in the monthly payments for these two loans.

A rectangular room has length L and width W, where L and W are measured in feet. a. Express the area in square feet algebraically. b. If carpeting costs x dollars per square yard, express the cost of carpeting this room algebraically.

A gazebo in the shape of a regular octagon has equal sides of 9 feet and an apothem of 10.9 feet. a. If one side of a gazebo is open, and the other sides have a railing, fi nd the cost of the railing if it sells for \(7.90 per foot. b. Find the area of the gazebo. c. Find the cost of the gazebo鈥檚 fl oor if the fl ooring costs \)3 per square foot. Round to the nearest hundred dollars.

Helene and Vick moved into an apartment and pay \(\$ 1,875\) rent per month. The landlord told them them the rent has increased 3.28\(\%\) per year on average. Express the rent \(y\) as an exponential function of the number of years they rent the apartment.
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