Compound Interest with Quarterly Compounding – 6 months at 20% p.a.: A moneylender lends ₹ 2,000 for 6 months at a nominal 20% per annum, with interest compounded quarterly. What amount will he receive at the end of the period?

Introduction / Context:
Nominal annual rates split across compounding subperiods must be converted to a per-period rate before exponentiation. Six months with quarterly compounding equals two quarter-periods at one-quarter of the nominal annual rate each period.

Given Data / Assumptions:

• P = 2,000
• Nominal r = 20% per annum
• Compounded quarterly ⇒ periodic rate = 20%/4 = 5% per quarter
• Time = 6 months = 2 quarters

Concept / Approach:
Use A = P * (1 + i)^n with i = 0.05 and n = 2. This captures the interest-on-interest across the two compounding quarters.

Step-by-Step Solution:
Periodic rate i = 0.20 / 4 = 0.05Number of periods n = 2A = 2,000 * (1.05)^2 = 2,000 * 1.1025 = 2,205

Verification / Alternative check:
Quarter 1: 2,000 → 2,100. Quarter 2: 2,100 * 1.05 = 2,205. Both methods agree on the final amount.

Why Other Options Are Wrong:
₹ 2,200 ignores the second compounding step; ₹ 2,160 and ₹ 2,040 correspond to lower effective rates or simple interest; ₹ 2,250 would require 12.5% per quarter or longer time.

Common Pitfalls:
Applying 20% annually for half a year as 10% simple interest misses the quarterly compounding structure, which yields a slightly higher amount than ₹ 2,200.

Final Answer:
₹ 2,205