Compound Interest with Half-Yearly Compounding – Find time from amount: ₹ 1,600 at 10% per annum compounded half-yearly amounts to ₹ 1,944.81 in how much time?

Introduction / Context:
When interest is compounded half-yearly, the annual rate is split across two periods per year. Determining time then reduces to counting the number of half-year periods needed to reach the given amount using the periodic factor.

Given Data / Assumptions:

• P = 1,600
• Nominal r = 10% per annum
• Compounded half-yearly ⇒ periodic rate = 10% / 2 = 5%
• Target amount A = 1,944.81

Concept / Approach:
With half-yearly compounding, A = P * (1 + 0.05)^n, where n is the number of half-year periods. Solve (1.05)^n = A / P = 1,944.81 / 1,600 = 1.215506… Recognize that (1.05)^4 = 1.21550625, implying n = 4 half-years ⇒ t = 2 years.

Step-by-Step Solution:
Compute ratio: A/P = 1,944.81 / 1,600 ≈ 1.215506Check powers: (1.05)^2 = 1.1025; (1.05)^4 = 1.21550625Thus n = 4 half-years ⇒ time t = 4 / 2 = 2 years

Verification / Alternative check:
Forward compute: A = 1,600 * (1.05)^4 = 1,600 * 1.21550625 = 1,944.81 (exact to paise), confirming 2 years.

Why Other Options Are Wrong:
1.5, 2.5, or 3 years correspond to 3, 5, or 6 half-year periods respectively, which yield amounts different from ₹ 1,944.81.

Common Pitfalls:
Using 10% as the per-period rate would double-count; for half-yearly compounding, use 5% per half-year and count half-year periods.

Final Answer:
2 years