The rate of compound interest is 60% per half-year. What is the minimum number of half-years in which a sum of money, invested at this rate of compound interest, will become four times (quadruple) its original value?

Introduction / Context:
This is a growth factor problem in compound interest. You are given a very high periodic rate (60% per half-year), and you must determine how many such periods are needed for the principal to become four times its original amount. Such problems can be solved using exponential growth ideas and involve comparing (1 + rate)^n with the desired multiple.

Given Data / Assumptions:
- Periodic rate of interest = 60% per half-year, compounded each half-year.
- Let the initial principal be P rupees.
- We want the amount to become 4P (four times the principal).
- We must find the minimum number of half-years n such that the amount ≥ 4P.

Concept / Approach:
Under compound interest, the amount after n periods is A = P * (1 + r)^n, where r is the periodic rate as a decimal. Here r = 0.60, so the growth factor per half-year is 1.6. We need the smallest integer n such that P * (1.6)^n ≥ 4P. Cancelling P from both sides, we get (1.6)^n ≥ 4, and we test integer values of n until this inequality holds.

Step-by-Step Solution:
Step 1: Write the compound amount formula: A = P * (1.6)^n, since 1 + 0.60 = 1.6.Step 2: We require A ≥ 4P.Step 3: So P * (1.6)^n ≥ 4P.Step 4: Cancel P (assuming P > 0): (1.6)^n ≥ 4.Step 5: Test n = 1: (1.6)^1 = 1.6 (less than 4).Step 6: Test n = 2: (1.6)^2 = 2.56 (still less than 4).Step 7: Test n = 3: (1.6)^3 = 4.096 (which is greater than 4).Step 8: Since for n = 2 we are below 4 and for n = 3 we exceed 4, the minimum number of half-years required is 3.

Verification / Alternative check:
If P = Rs. 1, then after 3 half-years at 60% per half-year, the amount is 1 * 1.6^3 = 4.096 ≈ 4.1, slightly more than four times. After 2 half-years, the amount would be only 2.56, which is less than 4. Thus, indeed 3 half-years is the earliest time when the sum quadruples or more.

Why Other Options Are Wrong:
2 half-years gives only 2.56 times the original value, not four times. 4, 5 or 6 half-years also quadruple the sum but are not minimal, since the question specifically asks for the minimum number of half-years. Hence only 3 half-years satisfies both becoming at least four times and being minimal.

Common Pitfalls:
Students sometimes misinterpret the rate as 60% per annum instead of per half-year or use simple interest logic. Another common error is to assume doubling twice (2 * 2 = 4) and equate that with 2 periods, ignoring that each period increases by 60%, not 100%. Always work directly with the compound factor (1 + r)^n and compare it numerically to the target multiple.

Final Answer:
The minimum number of half-years required is 3 half-years.