The compound interest on Rs. 24,000 invested at 10% per annum for 1.5 years, when interest is compounded semi-annually, is equal to how much amount (in rupees)?

Introduction:
This problem tests understanding of compound interest with a compounding period that is not yearly. The investment is for one and a half years, and the interest is compounded semi-annually, meaning twice per year. You must correctly convert the annual rate into a half yearly rate and use the right number of compounding periods.

Given Data / Assumptions:

• Principal (P) = Rs. 24,000
• Annual nominal rate (r) = 10% per annum
• Compounding frequency = semi-annual (twice a year)
• Total time = 1.5 years

Concept / Approach:
If interest is compounded semi-annually, we divide the annual rate by 2 and multiply the number of years by 2 to find the number of periods. The amount after n periods is given by:
A = P * (1 + i)^nwhere i is the rate per period and n is the number of periods. The compound interest CI is then:
CI = A - P

Step-by-Step Solution:
Step 1: Compute the half yearly rate: i = r/2 = 10% / 2 = 5% = 0.05 per half year.Step 2: Compute the number of half yearly periods: n = 1.5 years * 2 = 3 periods.Step 3: Use the formula A = P * (1 + i)^n.Step 4: Substitute values: A = 24,000 * (1 + 0.05)^3.Step 5: Calculate the growth factor: (1.05)^3 = 1.05 * 1.05 * 1.05 = 1.157625.Step 6: Multiply by the principal: A = 24,000 * 1.157625 = 27,783.Step 7: Find compound interest: CI = A - P = 27,783 - 24,000 = 3,783.Thus, the compound interest is Rs. 3,783.

Verification / Alternative check:
You can approximately check by noting that for 1 year at 10%, simple interest would be Rs. 2,400. For half a year, simple interest would be around Rs. 1,200. Adding a small extra due to compounding suggests a figure slightly above Rs. 3,600. The value Rs. 3,783 fits this expectation well.

Why Other Options Are Wrong:
Rs. 3,600 and Rs. 3,720 are closer to simple interest approximations and ignore the exact compounding over three half yearly periods. Rs. 3,840 and Rs. 3,900 are overestimates resulting from using an incorrect rate or wrong number of compounding periods.

Common Pitfalls:
A frequent mistake is to treat 1.5 years as 1 year or 2 years, or to apply 10% directly for 1.5 years without adjusting for semi annual compounding. Another common error is incorrectly computing (1.05)^3. Always ensure that both the rate per period and number of periods are consistent with the compounding frequency.

Final Answer:
The compound interest on Rs. 24,000 at 10% per annum, compounded semi annually for 1.5 years, is Rs. 3,783.