At 6% per annum interest compounded half-yearly, in what time will Rs. 3300 amount to Rs. 3399?

Introduction / Context:
This question tests understanding of compound interest when the interest is compounded half-yearly. The increase in amount over the principal is relatively small, so the time period is likely to be a fraction of a year. Identifying the correct compounding period and rate is the key step.

Given Data / Assumptions:

• Principal P = Rs. 3300
• Final amount A = Rs. 3399
• Nominal annual rate r = 6% per annum
• Compounding half-yearly, that is twice per year
• We need to find the time T in years

Concept / Approach:
For nominal rate r compounded half-yearly, the period rate i is:
i = r / 2 = 6% / 2 = 3% per half-yearThe compound amount after n half-yearly periods is:
A = P * (1 + i / 100)^nWe are given both A and P and must find n, then convert n to years by noting that 2 periods correspond to 1 year.

Step-by-Step Solution:
Step 1: Use the compound amount formula with half-yearly compounding.3399 = 3300 * (1.03)^nStep 2: Divide both sides by 3300.3399 / 3300 = (1.03)^n3399 / 3300 = 1.03Therefore (1.03)^n = 1.03Step 3: Solve for n.n = 1 half-yearly periodStep 4: Convert to years.Since 1 period corresponds to half a year, T = 0.5 year = half year

Verification / Alternative check:
Directly check one half-year of compounding: Amount after one half-year:
A = 3300 * (1 + 3 / 100) = 3300 * 1.03 = 3399This exactly matches the required amount, confirming that the time is one half-year.

Why Other Options Are Wrong:
Quarter year or 1.5 years: These would correspond to partial or multiple periods and would not produce exactly Rs. 3399.1 year or 2 years: These correspond to 2 or 4 half-yearly periods, which would give significantly larger amounts than Rs. 3399.

Common Pitfalls:
Students may forget that the quoted 6% is per annum and that for half-yearly compounding, they must use 3% per half-year. Others confuse the number of periods with the number of years. Always relate the number of compounding periods to the given compounding frequency to interpret the time correctly.

Final Answer:
The required time for Rs. 3300 to amount to Rs. 3399 at 6% per annum compounded half-yearly is half year.