Problem 19 - Algebra and Trigonometry

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?

Step-by-Step Solution

Answer

Dr. Gupta can afford a loan of approximately \(\$583,800\).

Step 1: Understanding the Problem

Dr. Gupta wants to know how much she can borrow given a monthly payment, interest rate, and loan length. The formula to use is for monthly mortgage payments.

Step 2: Set Up the Formula

The mortgage payment formula is: \[P = \frac{M \times (1 - (1 + r)^{-n})}{r}\]where \( P \) is the principal loan amount, \( M \) is the monthly payment, \( r \) is the monthly interest rate, and \( n \) is the total number of payments.

Step 3: Convert Terms to Formula Compatible Units

The annual interest rate is 6\(\%\), so the monthly interest rate \( r \) is \(\frac{6}{100 \times 12}\). The loan term is 30 years, so the number of monthly payments \( n \) is \(30 \times 12\).

Step 4: Calculate the Monthly Interest Rate

Convert the annual interest rate to a monthly rate: \[r = \frac{0.06}{12} = 0.005\]

Step 5: Calculate Total Number of Payments

Calculate the total number of payments for the 30-year loan: \[n = 30 \times 12 = 360\]

Step 6: Plug Values into the Formula

Substitute the known values into the mortgage payment formula: \[P = \frac{3500 \times (1 - (1 + 0.005)^{-360})}{0.005}\]

Step 7: Perform the Calculations

First, calculate the term inside the parentheses: \[(1 + 0.005)^{-360} \approx 0.1660\]This gives \[P = \frac{3500 \times (1 - 0.1660)}{0.005} \approx \frac{3500 \times 0.8340}{0.005}\] \ Solve for \(P\): \[P \approx \frac{2919}{0.005} = 583,800\]

Step 8: Interpretation

Dr. Gupta can afford a loan of approximately \(\$583,800\) with her monthly payment and interest rate.

Key Concepts

Interest Rate

Interest rate is a crucial part of mortgage calculations. It represents the cost of borrowing money. In simple terms, when a bank loans you money, they charge interest for lending it to you. This interest is usually given as an annual percentage rate (APR), which shows what percentage of the loan amount you need to pay back each year as interest.
In the case of Dr. Gupta's mortgage, the interest rate is 6\(\%\), which is an annual rate. To calculate the monthly interest rate, we need to divide the annual rate by 12 months. This gives us 0.5\(\%\) per month, or in decimal form:

• Monthly Interest Rate, \(r\) = \( \frac{6}{100 \times 12} = 0.005\)

Understanding how interest rates work helps predict how much you will owe over the lifetime of the loan. Lower rates mean less you have to pay in interest, lowering the total cost of the loan. It is always good to shop around for the lowest rates to save money in the long run.

Monthly Payment

Monthly payment is the fixed amount paid by the borrower each month to repay the loan. It includes both interest and a portion of the principal loan amount. For a mortgage, this is often set at a consistent rate for the life of the loan, making budgeting easier.

• In Dr. Gupta's situation, her monthly payment is \(\$3500\).

This payment is crucial in determining how much you can afford to borrow. Higher monthly payments allow you to borrow more since you're paying off both the loan and the interest quicker. However, it's also essential to ensure the monthly payment fits within your budget.
To calculate what loan size is affordable, the monthly payment amount is used alongside the interest rate and the loan term in the mortgage payment formula.

Loan Term

The loan term is the amount of time over which you will repay the loan. It is usually presented in years, and common terms for mortgages range from 15 to 30 years. The loan term influences both the amount you can borrow and your monthly payment.
A longer loan term typically means:

• Lower monthly payments because the repayment is spread over many years.
• More interest paid over the life of the loan since interest accumulates over a more extended period.

In Dr. Gupta's example, the loan term is 30 years. This means that she will make 360 monthly payments (since 30 years \(\times\) 12 months = 360 payments). It is important to choose a loan term that balances comfortable monthly payments with total interest paid.

Principal Loan Amount

The principal loan amount is the total amount borrowed at the beginning of the mortgage. It is the initial amount you receive from the lender, not including interest. This is essentially the part of the mortgage that needs to be paid back, with interest being the cost of borrowing it.
To calculate the maximum principal loan amount Dr. Gupta can afford, we use a specific formula taking into account her monthly payment, interest rate, and total number of payments.

• Using the mortgage payment formula, she can afford a principal amount, \(P\), approximated as \(\$583,800\).

The principal amount represents the "purchasing power" Dr. Gupta has in buying a home. Managing the principal amount wisely along with being aware of interest implications can significantly impact the overall cost of the mortgage.