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

Let the principal be ₹ P and rate of interest be r%.

Now, $$13920=P+\dfrac{P\times\ r\times\ 3}{100}$$

or, $$13920-P=\dfrac{P\times\ r\times\ 3}{100}$$ ---->(1)

Also, $$18960=P+\dfrac{P\times\ r\times\ 13}{100\times\ 2}$$

$$18960-P=\dfrac{P\times\ r\times13}{100\times\ 2}$$ ----->(2)

Dividing eqn(1) by eqn(2),

$$\dfrac{13920-P}{18960-P}=\dfrac{3}{\frac{12}{2}}=\dfrac{6}{13}$$

or, $$\left(13920-P\right)13=\left(18960-P\right)6$$

or, $$13920\times\ 13-13P=18960\times\ 6-6P$$

or, $$13920\times\ 13-18960\times\ 6=13P-6P$$

or, $$180960-113760=7P$$

or, $$67200=7P$$

or, $$P=\dfrac{67200}{7}=9600$$

Putting this in equation (1),

$$13920-9600=\dfrac{9600\times\ r\times\ 3}{100}$$

or, $$4320=96\times\ 3r$$

or, $$r=\dfrac{4320}{96\times\ 3}=15$$

So, rate percent is $$15\%$$

Now if the same sum had been invested for 2 years at the same rate as before but with interest compounded every 6 months, amount = $$9600\left(1+\dfrac{\frac{15}{2}}{100}\right)^4=9600\left(1+\frac{7.5}{100}\right)^4$$

So, interest = $$9600\left(1+\frac{7.5}{100}\right)^4-9600$$

= Rs 3220.50

= Rs 3221

So, the total interest earned is Rs 3221.