In how many years will a sum of Rs. 800 at 10% per annum, compounded semi-annually, amount to Rs. 926.10?

Introduction / Context:
This problem involves compound interest with semi annual compounding. We are given the principal, the annual nominal rate, the compounding frequency, and the final amount, and must determine the time required. This type of question helps learners practice the compound interest formula when interest is applied more than once per year.

Given Data / Assumptions:

Principal P = Rs. 800.
Nominal annual interest rate = 10% per annum.
Interest is compounded semi annually, so there are 2 compounding periods per year.
Final amount A = Rs. 926.10.
We must find the time in years for the amount to reach 926.10.

Concept / Approach:
For compound interest with m compounding periods per year, the amount after n compounding periods is A = P * (1 + r / m)^n, where r is the annual rate in decimal form. Here m = 2 and r = 0.10, so the periodic rate is 0.10 / 2 = 0.05 or 5 percent per half year. We set up the equation 926.10 = 800 * (1.05)^n and solve for n. Then time in years is n divided by 2, because there are 2 periods per year.

Step-by-Step Solution:
Principal P = 800, final amount A = 926.10. Annual rate r = 10% = 0.10 in decimal. Semi annual compounding means 2 periods per year, so periodic rate = 0.10 / 2 = 0.05. Let n be the number of half yearly periods. Compound interest formula: A = P * (1 + periodic rate)^n. So 926.10 = 800 * (1.05)^n. Divide both sides by 800: (1.05)^n = 926.10 / 800 = 1.157625. We observe that (1.05)^3 = 1.05 * 1.05 * 1.05 = 1.157625. Thus n = 3 half yearly periods. Time in years = n / 2 = 3 / 2 = 1.5 years.

Verification / Alternative check:
Check by forward calculation. After one half year, amount = 800 * 1.05 = 840. After two half years (one full year), amount = 840 * 1.05 = 882. After three half years, amount = 882 * 1.05 = 926.10. This matches the given final amount exactly, confirming that the required time is 1.5 years.

Why Other Options Are Wrong:
Option 2.5 years would correspond to 5 half years, and (1.05)^5 is greater than 1.276, which would produce an amount much larger than 926.10.
Option 3.5 years means 7 half years, leading to a still larger amount above 1,000 when starting from 800 at this rate.
Option 4.5 years means 9 half years, which would overshoot the target by an even greater margin under compound interest.

Common Pitfalls:
Some learners mistakenly apply simple interest instead of compound interest, which does not produce the correct progressive multiplication by 1.05 each half year.
Others use 10% as the rate per half year instead of dividing by 2, effectively compounding at 20% per year, which greatly changes the result.
A common algebraic mistake is to attempt to solve for n without realising that inspection of powers of 1.05 can quickly show that n = 3 works exactly.

Final Answer:
The sum will amount to Rs. 926.10 in 1.5 years when compounded semi annually at 10 percent per annum.