Kramer borrows 4000 dollars from George at an interest rate of 7% per annum compounded semiannually. The loan will be repaid by three payments: 1000 dollars due two years after the loan, a second payment due three years after the loan, and a third payment due five years after the loan. If the second payment is to be twice the size of the third payment, what is the amount of the third payment (in dollars)?

Introduction / Context:
This question involves loan repayment under compound interest with multiple payments at different times. It tests understanding of present value calculations, time value of money, and how to solve for unknown payment amounts when a certain relationship between payments is specified, in this case that one payment is twice another.

Given Data / Assumptions:

• Loan principal at time zero: 4000 dollars.
• Nominal annual interest rate j = 7% per annum.
• Interest is compounded semiannually, so there are 2 compounding periods per year.
• Periodic interest rate i = 0.07 / 2 per half year.
• First payment: 1000 dollars at the end of year 2.
• Second payment: amount 2x at the end of year 3.
• Third payment: amount x at the end of year 5.

Concept / Approach:
We equate the present value of all future payments to the original loan amount, discounted using the semiannual interest rate. Then we solve for x, the amount of the third payment. Since 1 year corresponds to 2 half year periods, year 2 is 4 periods, year 3 is 6 periods, and year 5 is 10 periods. We discount each payment back to time zero and set the sum equal to 4000.

Step-by-Step Solution:
Step 1: Compute the periodic interest rate i = 0.07 / 2 = 0.035 per half year.Step 2: Express the present value PV of each payment.Step 3: First payment PV: 1000 / (1 + i)^4, because 2 years is 4 half years.Step 4: Second payment PV: (2x) / (1 + i)^6, because 3 years is 6 half years.Step 5: Third payment PV: x / (1 + i)^10, because 5 years is 10 half years.Step 6: Set up the equation: 4000 = 1000 / (1 + i)^4 + (2x) / (1 + i)^6 + x / (1 + i)^10 and solve for x.Step 7: Using numerical calculation with i = 0.035, this gives x approximately equal to 1339.33 dollars.

Verification / Alternative check:
Once we have x, we can compute 2x and then verify that the total present value of 1000 at year 2, 2x at year 3, and x at year 5 at a semiannual rate of 3.5% equals roughly 4000. Small rounding differences may occur, but the equality should hold to within a few cents. This confirms that the value x = 1339.33 is consistent with the original loan amount and the required payment structure.

Why Other Options Are Wrong:

• 1389.00: This value makes the present value of the payments exceed the loan amount or fall short once you enforce the 2x relationship, so the equation does not balance.
• 1359.00: This is also not the root of the present value equation and leads to an incorrect total when discounted.
• 1379.00: This is close but still does not satisfy the exact present value equality at 7% compounded semiannually.

Common Pitfalls:
Common errors include treating the annual rate as the per period rate without dividing by two, using years instead of half year periods in the exponents, or discounting payments incorrectly. Some students also forget to use the relationship that the second payment is twice the third payment, which reduces the number of unknowns. Algebraic manipulation can be complex here, so it is important to keep track of exponents and to use a clear step by step approach or a calculator for the final numerical solution.

Final Answer:
The amount of the third payment, given that the second payment is twice as large, is approximately 1339.33 dollars.