Problem 23 - Precalculus: Mathematics for Calculus

Question

Mortgage Dr. Gupta is considering a 30 -year mortgage at \(6 \%\) interest. She can make payments of \(\$ 3500\) a month. What size loan can she afford? Mortgage \(A\) couple can afford to make a monthly mortgage payment of \(\$ 650 .\) If the mortgage rate is \(9 \%\) and the couple intends to secure a 30 -year mortgage, how much can they borrow? Financing a Car Jane agrees to buy a car for a down payment of \(\$ 2000\) and payments of \(\$ 220\) per month for 3 years. If the interest rate is \(8 \%\) per year, compounded monthly, what is the actual purchase price of her car? Financing a Ring Mike buys a ring for his fiancee by paying \(\$ 30\) a month for one year. If the interest rate is \(10 \%\) per year, compounded monthly, what is the price of the ring? Mortgage \(\quad\) A couple secures a 30 -year loan of \(\$ 100,000\) at \(9 \frac{3}{4} \%\) per year, compounded monthly, to buy a house. (a) What is the amount of their monthly payment? (b) What total amount will they pay over the 30 -year period? (c) If, instead of taking the loan, the couple deposits the monthly payments in an account that pays \(9 \frac{3}{4} \%\) interest per year, compounded monthly, how much will be in the account at the end of the 30 -year period?

Step-by-Step Solution

Answer

(a) $851.62, (b) $306,583.20, (c) $2,122,662.41.

Step 1: Understanding the Formula

The monthly payment for a mortgage can be determined using the formula \( M = P \frac{r(1+r)^n}{(1+r)^n-1} \), where \( M \) is the monthly payment, \( P \) is the principal loan amount, \( r \) is the monthly interest rate (annual rate divided by 12), and \( n \) is the total number of payments.

Step 2: Identifying Given Values

For part (a), we know the principal \( P = 100,000 \), annual interest \( 9.75\% \), and the loan term of \(30\) years. The monthly interest rate \( r = \frac{9.75\%}{12} = 0.8125\% \) or \( 0.008125 \). The total payments \( n = 360 \).

Step 3: Calculating the Monthly Payment

Substitute the known values into the monthly payment formula: \( M = 100,000 \frac{0.008125(1+0.008125)^{360}}{(1+0.008125)^{360} - 1} \). Calculating this gives a monthly payment \( M = 851.62 \).

Step 4: Calculating Total Payment Over the Loan's Life

For part (b), calculate the total amount paid over 30 years by multiplying the monthly payment by the number of months: \( 851.62 \times 360 = 306,583.20 \).

Step 5: Comparative Future Value Calculation

For part (c), the future value of the series of monthly payments can be calculated using \( FV = P \frac{(1+r)^n - 1}{r} \). Substitute \( P = 851.62, r = 0.008125, n = 360 \) to get \( FV = 2,122,662.41 \).

Key Concepts

Mortgage Calculation

When you are considering a mortgage, it's crucial to know how much you can borrow based on your desired monthly payment. This involves calculating the total loan amount, which depends on the mortgage term, interest rate, and the amount you wish to pay monthly. Mortgages are typically offered for terms of fixed years, like 15, 20, or 30 years. The overall interest paid can significantly impact the total costs. Longer terms may lower monthly payments but generally increase the total interest paid over time.

Before you apply for a mortgage, consider how the interest rate affects your borrowing limit. The higher the interest rate, the less you can afford to borrow for a given payment amount. Conversely, a lower interest rate increases the principal amount you can afford to borrow. Understanding these variables is crucial, as they directly affect the monthly expenses and the total cost of the home you can acquire.

Interest Rate and Compounding

Interest rates are a fundamental component of both borrowing and investing. They determine the cost of borrowing money and are often expressed as an annual percentage rate (APR). Compounding interest refers to the process where interest is added to the principal sum of a loan or deposit, so that from that moment on, the interest that has been added also earns interest.

Let's delve into how interest rates work. They are typically compounded regularly—be it annually, semi-annually, quarterly, or most commonly, monthly. When rates are compounded monthly, the total interest paid or received over time can be more than anticipated. The formula for converting an annual interest rate to a monthly one is simply to divide the annual rate by 12. For example, a 9.75% annual interest becomes 0.8125% per month, which is crucial for accurate loan calculations.

• Annual interest rates divided by the compounding periods determine the periodic interest rate.
• Monthly compounding results in more total interest than annual compounding.
• Understanding compounding helps you manage your loans and investments effectively.

Monthly Payment Formula

The monthly payment formula is a key tool for anyone considering a loan. It's used to calculate the amount you'll pay each month over the term of a loan. This formula considers the loan principal, the interest rate, and the number of payments. It allows you to determine the exact monthly payments, which means budgeting becomes far more straightforward.

Here’s how the formula looks: \[M = P \frac{r(1+r)^n}{(1+r)^n-1}\]where:

• \(M\) is the monthly payment;
• \(P\) is the principal amount;
• \(r\) is the monthly interest rate, calculated by dividing the annual rate by 12;
• \(n\) is the total number of payments (months in most cases).

Using this formula helps borrowers see the interplay between the principal, interest, and length of time. Remember, the more months you spread the repayments over, the less each installment will be, but this often results in more interest paid overall. Getting to grips with this formula will aid in planning sustainable and manageable payments.