Compound discount on a future sum: Find the compound-discount amount (i.e., the reduction from the sum due) on $5229 payable after 1 year 9 months at 5% per annum (compounded annually, fractional year allowed).

Introduction / Context:
Under compound interest, present worth PW = S / (1 + i)^t, where i is the effective annual rate and t is time in years. The compound “discount” asked here is the difference S − PW. Although the period includes a fractional year, we can apply the real-exponent formula directly.

Given Data / Assumptions:

• S = $5229 (sum due).
• i = 5% per annum (annual compounding).
• t = 1 year 9 months = 1.75 years.

Concept / Approach:
Compute PW using PW = S / (1.05)^1.75. Then the discount D = S − PW. Rounding to the nearest cent yields the closest option. Using a fractional exponent is standard for compound interest over non-integer years.

Step-by-Step Solution:

Verification / Alternative check:
Using a calculator with higher precision for 1.05^1.75 refines D to about $429, matching option (A). Small rounding differences may occur with logarithmic approximations.

Why Other Options Are Wrong:

• $415, $393.25: Too low; they correspond to using simple interest or an understated compounding effect.

Common Pitfalls:
Applying simple-interest discount instead of compound, or rounding the growth factor too early. Compute PW first, then subtract.

Final Answer:
$ 429.00