In how many years will a sum of ` 800 at 10% per annum compounded semi-annually become ` 926.10?

The correct option is 1 1/2 years.

Let's understand the step-by-step calculation to find the time period.

First, we identify the given values from the problem:
- Principal sum (P) = ₹ 800
- Amount (A) = ₹ 926.10
- Rate of interest (R) = 10% per annum

Since the interest is compounded semi-annually (every 6 months):
- The rate of interest for each half-year becomes half of the annual rate:
$$R_{\text{half-year}}=\frac{10\%}{2}=5\%$$
- If the time period is t years, the number of half-years (n) will be 2t.

The formula for the compound amount is:
$$A=P{(1+\frac{R}{100})}^n$$
Substituting the values:
$$926.10=800{(1+\frac{5}{100})}^n$$
Divide both sides by 800:
$$\frac{926.10}{800}={(1+\frac{1}{20})}^n$$
Simplify both sides:
$$\frac{92610}{80000}={\left(\frac{21}{20}\right)}^n$$
$$\frac{9261}{8000}={\left(\frac{21}{20}\right)}^n$$
Recognizing that 9261 is 21³ and 8000 is 20³:
$${\left(\frac{21}{20}\right)}^3={\left(\frac{21}{20}\right)}^n$$
Comparing the exponents, we get:
$$n=3\text{half-years}$$
Since there are 2 half-years in a full year, the time (t) is:
$$t=\frac{3}{2}=1\frac{1}{2}\text{years}$$