- Calculate the loan amount by subtracting the 20% down payment from the house price: \(L = 309900 \times 0.8 = 247920\).
- Determine the monthly interest rate (\(r=0.005\)) and the total number of payments (\(n=360\)).
- Apply the loan amortization formula: \(M = L \times \frac{r(1+r)^n}{(1+r)^n - 1}\).
- Calculate the monthly payment to be approximately \(\boxed{\$1,486.41}\) (Option c).
Explanation
- Analysis and Setup of the Loan Calculation
We are given the following data regarding a home loan:
- Purchase price of the house: \(P = \$309,900\).
- Down payment: \(20\%\) of the purchase price.
- Loan term: \(T = 30\) years.
- Annual interest rate: \(r_{annual} = 6\% = 0.06\).
The loan amount \(L\) is the total price minus the down payment:
\[L = P \times (1 - 0.20) = 309900 \times 0.80 = \$247,920\]
The monthly interest rate \(r\) is the annual rate divided by 12:
\[r = \frac{0.06}{12} = 0.005\]
The total number of monthly payments \(n\) over 30 years is:
\[n = 30 \times 12 = 360\]
We will use the standard amortization formula for monthly payments \(M\):
\[M = L \times \frac{r(1+r)^n}{(1+r)^n - 1}\]
- Calculation of the Monthly Payment
Now, we substitute our values into the amortization formula:
\[M = 247920 \times \frac{0.005 \times (1+0.005)^{360}}{(1+0.005)^{360} - 1}\]
First, calculate \((1+r)^n\):
\[(1+0.005)^{360} = (1.005)^{360} \approx 6.022575\]
Now, substitute this back into the formula:
\[M = 247920 \times \frac{0.005 \times 6.022575}{6.022575 - 1}\]
\[M = 247920 \times \frac{0.030112875}{5.022575} \approx 247920 \times 0.0059975\]
Performing the multiplication yields approximately \(1486.4057\). Rounding this to the nearest cent, we get:
\[M \approx \$1,486.41\]
Comparing this result to the given options, option (c) is \(\$1,486.41\).
- Conclusion
Based on our calculations, the monthly payment for a loan of \(\$247,920\) with a monthly interest rate of \(0.5\%\) over \(360\) months is \(\$1,486.41\). This matches option (c). The final answer is \(\boxed{\$1,486.41}\).
Examples
Calculating a monthly mortgage payment is a fundamental skill in personal finance. Understanding how interest rates, loan terms, and down payments impact your monthly commitment allows you to plan your budget effectively. For instance, if you were considering a different property or exploring how an extra down payment could reduce your monthly expenses, you would use this same mathematical model to evaluate your financial situation and make informed decisions about your future home ownership.