What is the minimum number of half years in which a sum of money kept at 60 percent compound interest per half year will be quadrupled (become four times its original value)?

Introduction / Context:
This question asks about the time required for a sum of money to become four times its original value under compound interest. The rate is 60 percent per half year, and we must express the time in half years. This type of problem focuses on exponential growth under high interest rates and requires working with powers of the growth factor until the desired multiple is reached.

Given Data / Assumptions:

• Initial principal is some amount P.
• Rate of interest = 60 percent per half year.
• Compounding frequency: each half year.
• We want the amount to become 4P, that is, quadrupled.
• We must find the minimum number of half years n such that the amount is at least 4P.

Concept / Approach:
Under compound interest with rate r per period and n periods, the amount is: A = P * (1 + r/100)^n Here, r = 60 per half year, so the growth factor per half year is 1.60. We seek the smallest integer n such that: P * (1.60)^n ≥ 4P Dividing both sides by P, we need: (1.60)^n ≥ 4 We then test small integer values of n to find the minimum one that satisfies this inequality.

Step-by-Step Solution:
We look for n such that (1.60)^n ≥ 4 Compute powers of 1.60: For n = 1: (1.60)^1 = 1.60 (less than 4) For n = 2: (1.60)^2 = 2.56 (still less than 4) For n = 3: (1.60)^3 = 1.60 * 2.56 = 4.096 At n = 3, (1.60)^3 = 4.096, which is greater than 4 Thus after 3 half years, the amount becomes more than four times the original Therefore, the minimum number of half years required is 3

Verification / Alternative check:
We can express the condition more algebraically. Taking natural or base 10 logarithms on both sides of (1.60)^n = 4 gives: n * log(1.60) = log(4) n = log(4) / log(1.60) Computing this gives a value slightly greater than 2.7, which confirms that n must be at least 3. Since n must be an integer number of half year periods, 3 is the minimum integer satisfying the requirement.

Why Other Options Are Wrong:
At n = 2 half years, the factor is 2.56, which is less than 4, so the amount is less than quadrupled. Values like 4, 5, or 6 are larger than necessary since the amount already exceeds four times at n = 3. The question asks for the minimum number of half years, so any number greater than 3 is not correct even though the amount would still be at least four times for those periods.

Common Pitfalls:
Some students misinterpret the rate as 60 percent per year instead of per half year, which leads to using the wrong growth factor. Others incorrectly approximate powers of 1.60 or guess an answer without checking systematically. Confusion between doubling, tripling, and quadrupling can also occur. Systematically computing successive powers and understanding that the rate is per half year helps avoid these errors.

Final Answer:
The minimum number of half years required for the sum to be quadrupled is 3 half years.