Compound interest with sub-annual compounding: A present sum of ₹100 is invested for one year at a half-yearly interest rate of 10% (i.e., 10% every 6 months). What will be the future amount at the end of one year?

Introduction / Context:
This problem checks understanding of compound interest with multiple compounding periods in a year. When the nominal rate is quoted per half-year (every 6 months), we must compound twice within one year. Many learners mistakenly apply simple interest or compound only once per year, leading to incorrect results.

Given Data / Assumptions:

• Principal P = ₹100 (present sum at time 0).
• Interest rate per half-year i = 10% = 0.10.
• Number of half-years in one year n = 2.
• No withdrawals or additional deposits.

Concept / Approach:
For compound interest with sub-annual compounding, use the standard formula: F = P * (1 + i)^n, where i is the interest rate per compounding period and n is the number of such periods. Here each period is half a year, so n = 2 within one year.

Step-by-Step Solution:
Identify i and n: i = 0.10 per half-year, n = 2 half-years.Apply the compound interest formula: F = 100 * (1 + 0.10)^2.Compute the factor: (1.10)^2 = 1.21.Multiply by principal: F = 100 * 1.21 = ₹121.

Verification / Alternative check:
You can compute period by period: after first 6 months, amount = 100 * 1.10 = ₹110. After second 6 months, reapply 10% on ₹110: 110 * 1.10 = ₹121. This matches the direct formula result.

Why Other Options Are Wrong:
₹110: This applies only one half-year of interest instead of two.₹97 and ₹91: These imply negative or incorrect rate applications.₹105: A common guess confusing 10% per year with some averaging; still incorrect.

Common Pitfalls:
Forgetting the number of compounding periods, using simple interest (F = P * (1 + i_total)) rather than compounding, or mixing nominal annual and per-period rates. Always align i with the period count n.

Final Answer:
₹121