So here is the problem:Luke is borrowing $\$10{,}000$ from the bank. The bank offers him a choice between two $10$-year payment plans:
Plan 1 Luke's debt accumulates $10\%$ annual interest which compounds quarterly. Luke pays off half his balance after $5$ years, and the rest at the end of the $10$ years.
Plan 2 Luke's debt accumulates $10\%$ annual interest which compounds annually. Luke pays off his full balance at the end of the $10$ years.
What is the (positive) difference between Luke's total payments under Plan 1 and his total payments under Plan 2? Round to the nearest dollar.
Now, the second plan's payment amount is $10,000 * (1 +\frac{1}{10})^{10}$ which is around $25937$.
In the first plan, he will pay off $\frac{1}{2} *10000 (1+\frac{1}{10})^{20}$ after five years. The remaining amount will continue to be compounded, which he pays off after another five years, which is $\frac{1}{2} * 10000(1+\frac{1}{10})^{40}$ which is around 259933. The difference between the plans is a lot. Where did I go wrong?
On calculating the first plan, $\frac 12*10000(1+\frac{1}{10})^{20}$ should be replaced by $10000(1+\frac{2.5}{100})^{20}$ because in the FIRST FIVE years the interest is being compounded quarterly on the complete amount. The interest rate is 10% per year and not 10% per quarter.