A sum of ₹ 12,000 amounts to ₹ 15,972 at a certain rate percent per annum in 1$\frac{1}{2}$ years, when the interest is compounded half-yearly. What will be the amount of the same sum in same time and with the same rate, if interest is compounded annually ?

The correct answer is Option 1: ₹ 15,840.

Step 1: Understand the given details
Principal ($P$) = ₹ 12,000
Amount ($A$) = ₹ 15,972
Time period ($T$) = 1$\frac{1}{2}$ years = 3 half-years.
Let the annual rate of interest be $R\%$ per annum. Since interest is compounded half-yearly, the rate per half-year is $\frac{R}{2}\%$, and the number of compounding periods is $n=3$.

Step 2: Find the rate of interest
Using the compound interest formula for half-yearly compounding:

$A=P{\left(1+\frac{R/2}{100}\right)}^n$

Substitute the given values into the formula:

$15,972=12,000{\left(1+\frac{R}{200}\right)}^3$

Divide both sides by 12,000:

$\frac{15,972}{12,000}={\left(1+\frac{R}{200}\right)}^3$

Simplifying the fraction by dividing numerator and denominator by 12:

$\frac{1,331}{1,000}={\left(1+\frac{R}{200}\right)}^3$

Taking the cube root on both sides:

$\frac{11}{10}=1+\frac{R}{200}$

$\frac{R}{200}=\frac{11}{10}-1=\frac{1}{10}$

$R=20\%$ per annum.

Step 3: Calculate the amount when interest is compounded annually
Time period = 1$\frac{1}{2}$ years = 1 full year + $\frac{1}{2}$ year.
Rate of interest = 20% per annum.
For the first full year, interest rate is 20%.
For the remaining half year, interest rate is half of the annual rate, which is 10%.

The total amount after 1$\frac{1}{2}$ years compounded annually is given by:

$A=P\times \left(1+\frac{20}{100}\right)\times \left(1+\frac{10}{100}\right)$

$A=12,000\times \frac{6}{5}\times \frac{11}{10}$

$A=12,000\times \frac{66}{50}=240\times 66=15,840$

Thus, the amount when interest is compounded annually is ₹ 15,840.