At a certain simple rate of interest, a given sum amounts to Rs.13920 in 3 years, and to Rs 18960 in 6 years and 6 months. If the same given sum had been invested for 2 years at the same rate as before but with interest compounded every 6 months, then the total interest earned, in rupees, would have been nearest to

The correct answer is 3221.

Let the principal sum be $P$ and the simple rate of interest be $R$% per annum.

We know that the amount after 3 years is Rs. 13920, and after 6.5 years (which is 6 years and 6 months) it is Rs. 18960.

The difference between the two amounts is the simple interest accumulated over the additional 3.5 years (since 6.5 - 3 = 3.5).

Simple interest for 3.5 years = Rs. 18960 - Rs. 13920 = Rs. 5040.

Therefore, the simple interest for 1 year = $\frac{5040}{3.5}=1440$.

Now, we can find the simple interest for 3 years, which is $3\times 1440=4320$.

Since the amount after 3 years is Rs. 13920, the principal $P$ = Amount - Interest = $13920-4320=9600$.

We can now find the rate of interest $R$:

$R=\frac{1440\times 100}{9600}=15\%$

Next, we need to calculate the compound interest on this same principal (Rs. 9600) for 2 years at the same rate of 15% per annum, compounded every 6 months (half-yearly).

For half-yearly compounding, the rate per period is $\frac{15}{2}=7.5\%$ and the number of periods is $2\times 2=4$ half-years.

The formula for the total amount with compound interest is $A=P{(1+\frac{r}{100})}^n$, where $r$ is the rate per period and $n$ is the number of periods.

Amount $A=9600{(1+\frac{7.5}{100})}^4=9600{\left(1.075\right)}^4$

Calculating ${\left(1.075\right)}^4$ gives approximately 1.335469.

Amount $A\approx 9600\times 1.335469\approx 12820.50$

The total compound interest earned is Amount - Principal: $12820.50-9600=3220.50$.

The nearest integer to 3220.50 is 3221.

Thus, the total interest earned is nearest to Rs. 3221.