Final answer:
To repay a debt of $8,000 at 7.5% annual effective compound interest with a payment of $X two years from now and $2X four years from now, the present values of the future payments are calculated and summed. The value of X is found to be approximately $3385.67.
Step-by-step explanation:
To find the value of X when a payment of $X two years from now and a payment of $2X four years from now repays a debt of $8,000 with 7.5% annual effective compound interest, we must calculate the present value of these future payments and set it equal to the initial debt of $8,000.
The formula for the present value of a future payment at compound interest is:
Present Value = Future Value / (1 + interest rate)time
We will apply this formula twice, once for the $X payment and once for the $2X payment:
- Present Value of first payment: X / (1.075)2
- Present Value of second payment: 2X / (1.075)4
We sum these two present values and set them equal to $8,000:
X / (1.075)2 + 2X / (1.075)4 = $8,000
Solving for X, we find that X is approximately $3385.67 when rounded to the nearest cent. This demonstrates how compound interest accrues more substantially than simple interest, as highlighted in the examples provided.