A company that purchases a piece of equipment by borrowing $300,000 for 9 years at an annual rate of 6%, compounded monthly, has monthly payments of $3,601.72. Find the unpaid balance on this loan after 1 year. Round your answer to the nearest cent.

Question

A company that purchases a piece of equipment by borrowing $300,000 for 9 years at an annual rate of 6%, compounded monthly, has monthly payments of $3,601.72. Find the unpaid balance on this loan after 1 year. Round your answer to the nearest cent. ​

Quizplus · Accepted Answer

To find the unpaid balance after 1 year, we first need to understand that the loan is being repaid in monthly installments, with interest compounded monthly. The formula for the monthly payment on a loan is derived from the annuity formula, but since the monthly payment amount is already given, we focus on how to calculate the remaining balance.The remaining balance of a loan after a certain number of payments can be calculated using the formula for the future value of an annuity. However, in this case, it's more straightforward to calculate the total amount paid in the first year and then apply the interest that has been compounding on the principal to find out how much is left.Given:- Principal, $P = $300,000$- Annual interest rate, $r = 6\%$ or $0.06$ when expressed as a decimal- Monthly interest rate, $r_m = \frac{0.06}{12} = 0.005$- Number of years, $t = 9$- Number of payments per year, $n = 12$- Monthly payment, $M = $3,601.72$Total payments made in the first year: $12 \times $3,601.72 = $43,220.64$To find the unpaid balance, we use the formula for the remaining balance on an amortizing loan, which can be complex due to the compounding interest and decreasing principal. However, a simpler approach to estimate the unpaid balance after 1 year involves calculating the interest paid and principal repaid in the first year and subtracting from the original loan amount. This approach, while straightforward, doesn't directly apply here due to the compounding interest's effect on the balance over each month.The precise calculation of the unpaid balance after 1 year involves using the amortization formula or a financial calculator to account for the monthly compounding and repayment schedule. Given the options provided and understanding that the calculation requires specific financial formulas or a calculator:The correct answer is found through the amortization calculation, which takes into account the monthly payments, the interest rate, and the time period. After 1 year of payments, the unpaid balance is calculated to be $274,073.67, which corresponds to option B. This calculation considers the reduction in principal from each payment and the interest that accrues on the decreasing principal balance over the 12 months.

A Company That Purchases a Piece of Equipment by Borrowing