A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits ₹ 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:

### Concept & Multiple Deposits within a Year When multiple deposits are made, calculate the interest for each deposit independently based on how long it stays in the bank before the end of the term. The rate given is 5% half-yearly (the text says "5% calculated on half-yearly basis", which usually implies 5% p.a., thus 2.5% half-yearly, BUT if it means 5% per half year, let's verify the math. Standard interpretation of such phrasing in Indian exams implies the annual rate is 5%, making the half-yearly rate 2.5%). ### Step-by-Step Solution * **Given:** Deposit 1 = ₹ 1600 on Jan 1. Deposit 2 = ₹ 1600 on July 1. Rate = 5% p.a., compounded half-yearly. * **Step 1:** Establish the half-yearly rate. $r = \frac{5\%}{2} = 2.5\%$ per half-year. * **Step 2:** Analyze the 1st January deposit. It remains in the bank for 2 half-years (Jan-Jun, Jul-Dec). Amount = $1600 \times (1 + 0.025)^2 = 1600 \times (1.025)^2 = 1600 \times 1.050625 = 1681$ Interest from 1st deposit = $1681 - 1600 = 81$ * **Step 3:** Analyze the 1st July deposit. It remains in the bank for 1 half-year (Jul-Dec). Amount = $1600 \times (1 + 0.025)^1 = 1600 \times 1.025 = 1640$ Interest from 2nd deposit = $1640 - 1600 = 40$ * **Step 4:** Calculate total interest gained. Total Interest = $81 + 40 = 121$ ### Exam Strategy & Shortcut Instead of heavy decimals, use fractions. $2.5\% = \frac{1}{40}$. First deposit interest: $1600 \times (\frac{41}{40})^2 - 1600 = 1600 \times \frac{1681}{1600} - 1600 = 1681 - 1600 = 81$. Second deposit interest: $1600 \times \frac{41}{40} - 1600 = 1640 - 1600 = 40$. $81 + 40 = 121$. Fractions cancel out perfectly. ### Common Pitfall Assuming "5% calculated on half-yearly basis" means 5% per half year. If it were 5% per half year, interest would be $1600(1.05)^2 - 1600 + 1600(1.05) - 1600 = 164 + 80 = 244$, which is not in the options. This confirms the rate is 5% p.a. ### Final Answer Therefore, the correct answer is **₹ 121**.