Fifteen equal semiannual payments are made into a sinking fund that earns 7% interest compounded semiannually so that 4,850 dollars will be in the fund at the end. What is the amount of each payment (rounded to the nearest cent)?

Introduction / Context:
This question deals with a sinking fund, which is a fund created by making regular payments so that a specific future amount is accumulated. The fund earns compound interest at a given periodic rate, and you must determine the size of each equal payment that results in a target future value.

Given Data / Assumptions:

• Target future value FV = 4,850 dollars.
• Nominal annual interest rate = 7% compounded semiannually.
• Semiannual rate i = 7% / 2 = 3.5% = 0.035.
• Number of semiannual payments n = 15.
• Payments are deposited at the end of each semiannual period (ordinary annuity).

Concept / Approach:
Future value of an ordinary annuity is:
FV = R * ((1 + i)^n - 1) / i Here FV and i and n are known, and we solve for the payment R:
R = FV * i / ((1 + i)^n - 1)

Step-by-Step Solution:
Step 1: Set i = 0.035 and n = 15. Step 2: Compute (1 + i)^n = (1.035)^15. Step 3: Find the numerator FV * i = 4850 * 0.035. Step 4: Compute the denominator (1.035^15 - 1). Step 5: Divide to get R ≈ 251.35 dollars. Thus, each semiannual deposit should be about 251.35 dollars.

Verification / Alternative Check:
You can calculate the future value using this payment amount: multiply 251.35 by the annuity factor ((1.035)^15 - 1) / 0.035. The result is very close to 4,850 dollars, confirming the correctness of the payment size.

Why Other Options Are Wrong:
Option B (245.45 dollars) and option C (235.87 dollars) are too small, leading to a future value less than 4,850 dollars.
Option D (251.00 dollars) is close but slightly under the more accurate rounded value and would accumulate a smaller amount than required.
Option E (260.00 dollars) would produce more than 4,850 dollars in the fund.

Common Pitfalls:
Students sometimes treat the 7% as the semiannual rate instead of the annual nominal rate, which doubles the correct periodic rate and drastically changes the answer. Another common error is to invert the formula and mistakenly solve for FV instead of R.

Final Answer:
Each semiannual sinking fund payment should be approximately 251.35 dollars.