Question

Joan borrowed $20,000.00 to buy a car. She repaid $4800.00 three months later and $9000.00 eight months later. After twelve months, she borrowed an additional $7000.00, and repaid $5800.00 after 15 months. She paid the entire balance, including the interest, after 24 months. Interest was 8% compounded monthly for the first year and 8.3% compounded monthly for the remaining time. What was the size of the final payment? The final payment is $ (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

Answer 1

The final payment calculation involves applying compound interest to Joan's car loan after each transaction she makes. The interest rates applied are 8% for the first year and 8.3% for the second year, compounded monthly. Each repayment she makes affects the principal, which then accrues compound interest until the next transaction.

Step-by-step explanation:

To determine the size of Joan's final payment, we need to account for each transaction and apply compound interest for the appropriate time periods at the given rates.

Initially, Joan borrows $20,000.00. Three months later, she repays $4,800, which reduces the principal. The remaining balance then accrues interest at 8% per annum, compounded monthly, for the next five months until she makes the second payment of $9,000. After twelve months from the initial loan, she borrows an additional $7,000, which means the new principal amount is the balance after the first year's compound interest plus $7,000. This new total accrues interest at 8.3% per annum, compounded monthly. Three months into the second year, she repays $5,800. Finally, the remaining balance accrues interest for the next nine months until the 24-month mark, at which point she makes the final payment.

Calculating the compounded principal after each transaction and applying the monthly compounded interest requires careful application of the compound interest formula: P(1 + r/n)^(nt), where P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

The calculation of the final payment would require step by step computation of the balance after each transaction, taking into account the specific compounded interest rate applicable to each period.