Kramer borrowed 4,000 dollars from George at 7% interest compounded semiannually. The loan will be repaid by three payments: 1,000 dollars due 2 years after the loan, and two more payments due 3 and 5 years after the loan, with the second payment being twice the third. What is the amount of the third payment?

Introduction / Context:
This is a multi step time value of money problem involving unequal payments on a loan with semiannual compounding. You know the timing of the payments, the interest rate, and the relationship between the second and third payments, and you must determine the size of the final payment that makes the present value of all payments equal to the loan amount.

Given Data / Assumptions:

• Loan principal PV = 4,000 dollars at time 0.
• Nominal annual interest rate = 7% compounded semiannually.
• Semiannual rate i = 7% / 2 = 3.5% = 0.035.
• First payment: 1,000 dollars at t = 2 years.
• Second payment: P2 at t = 3 years.
• Third payment: P3 at t = 5 years.
• Relationship: P2 = 2 * P3.

Concept / Approach:
Convert each payment to its present value at time 0 using the semiannual rate and appropriate number of periods. Then set the sum of these present values equal to the loan principal and solve for the unknown third payment P3.
PV = 1000 / (1 + i)^4 + P2 / (1 + i)^6 + P3 / (1 + i)^10 with P2 = 2 * P3.

Step-by-Step Solution:
Step 1: Determine the number of semiannual periods. Two years correspond to 4 periods, three years to 6 periods, and five years to 10 periods.
Step 2: Substitute P2 = 2 * P3 = 2x and P3 = x into the present value equation: 4000 = 1000 / (1.035)^4 + 2x / (1.035)^6 + x / (1.035)^10. Step 3: Compute the discount factors 1 / (1.035)^4, 1 / (1.035)^6, and 1 / (1.035)^10. Step 4: Combine the terms in x and solve the resulting linear equation for x. Step 5: This yields x ≈ 1,339.33 dollars. So the third payment P3 is approximately 1,339.33 dollars, and the second payment P2 is about 2,678.66 dollars.

Verification / Alternative Check:
Substitute P3 ≈ 1,339.33 and P2 ≈ 2,678.66 back into the present value equation and recompute the total PV. The result will be very close to 4,000 dollars, confirming that these payment amounts are consistent with the loan terms.

Why Other Options Are Wrong:
Options A, B, and C (1,389.00; 1,359.00; 1,379.00 dollars) are larger than the correct value and would lead to a present value of payments that exceeds 4,000 dollars.
Option E (1,300.00 dollars) is smaller than the correct value and would not fully amortize the loan at the given interest rate.

Common Pitfalls:
Learners may mistakenly use annual periods instead of semiannual periods, which changes the exponents and gives an incorrect result. Others might forget that the second payment is twice the third and treat them as independent unknowns, making the algebra more complicated than necessary.

Final Answer:
The amount of the third payment that satisfies the loan conditions is approximately 1,339.33 dollars.