Find the compound interest on ₹12,000 at 8% per annum for 2 years 6 months, compounded annually.

The correct answer is ₹2,556.67.

To find the compound interest on a sum when the time period involves a fraction of a year (here, 2 years 6 months) and interest is compounded annually, we calculate the amount at the end of the full years first, and then apply simple interest for the remaining fractional year.

Given Data:

Principal (P) = ₹12,000
Rate of interest (R) = 8% per annum
Time (n) = 2 years 6 months = 2 $\frac{1}{2}$ years = 2.5 years

Step 1: Calculate the amount at the end of 2 full years.

The formula for the total amount $A$ under annual compounding is:

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

For the first 2 full years ($n=2$):

$A_2=12000\times {\left(1+\frac{8}{100}\right)}^2$

$A_2=12000\times {\left(1.08\right)}^2$

$A_2=12000\times 1.1664=₹\mathrm{13,996.80}$

Step 2: Calculate the interest for the remaining 6 months (0.5 years).

The amount at the end of 2 years (₹13,996.80) becomes the principal for the next 6 months ($\frac{1}{2}$ year).

Interest for the remaining $\frac{1}{2}$ year:

$\text{Interest for 6 months}=\frac{13996.80\times 8\times 0.5}{100}$

$\text{Interest for 6 months}=\frac{559.872}{1}=₹559.872$

Step 3: Calculate the total amount after 2 years 6 months.

$\text{Total Amount (A)}=13996.80+559.872=₹\mathrm{14,556.672}$

Step 4: Calculate Compound Interest.

$\text{Compound Interest (CI)}=\text{Total Amount}-\text{Principal}$

$\text{CI}=14556.672-12000=₹\mathrm{2,556.672}$

Rounding to two decimal places, the compound interest is ₹2,556.67.