In what time will Rs 3300 become Rs 3399 at 6% per annum interest compounded half yearly?

Introduction / Context:
This question asks for the time required for a small increase in principal under compound interest with half yearly compounding. The principal and final amount differ by only 99 rupees, suggesting a very short time interval. Understanding how the half yearly rate relates to the annual rate and how many compounding periods are needed to reach the new amount is key to solving this quickly.

Given Data / Assumptions:

• Principal P = Rs 3300
• Amount A = Rs 3399
• Nominal annual interest rate r = 6% per annum
• Interest is compounded half yearly
• We must find the time T in years

Concept / Approach:
With half yearly compounding, the periodic rate per half year is r / 2, and the number of periods is twice the number of years. The compound amount after n half year periods is A = P * (1 + r / 2)^n. Here, r = 6%, so the half yearly rate is 3%. If we notice that the amount Rs 3399 is exactly 3% more than Rs 3300, we can recognize that this corresponds to one half year of interest, because 3% of 3300 is 99. Therefore, the increase from 3300 to 3399 happens in just one half year period.

Step-by-Step Solution:
Annual rate r = 6%, so half yearly rate = r / 2 = 3% = 0.03 Check the amount after one half year: Interest for one half year = P * 0.03 = 3300 * 0.03 = 99 Amount after one half year = 3300 + 99 = 3399 This matches the given amount A Number of half year periods n needed = 1 Time in years T = n / 2 = 1 / 2 = 0.5 year

Verification / Alternative check:
We can also check with the general formula. A = P * (1 + r / 2)^n. Substituting n = 1 gives A = 3300 * (1 + 0.03) = 3300 * 1.03 = 3399. This is exactly the given amount, confirming that one compounding period of half a year is sufficient. Any additional periods would raise the amount further and not match the question data.

Why Other Options Are Wrong:
A quarter of a year would correspond to only half of the half yearly period, which is not how the compounding is defined. One year corresponds to two half year periods and would produce an amount larger than 3399. Two years correspond to four half year periods and would increase the amount even more. Only half a year gives the precise jump from 3300 to 3399 at a half yearly rate of 3%.

Common Pitfalls:
Some students mistakenly apply the 6% annual rate directly for a half year without dividing by two, leading to an incorrect step of 6% growth instead of 3%. Others attempt to solve for time using logarithms without noticing the simple proportional relationship between 3300 and 3399. Recognizing that the increase is exactly one half yearly interest payment allows a quick and accurate answer.

Final Answer:
Rs 3300 will become Rs 3399 in half a year, that is one half year compounding period at 6 percent per annum compounded half yearly.