The difference between compound and simple interests on a certain sum of money at the interest rate of 10 % per annum for $$1\frac{1}{2}$$ years is ₹183, when the interest is compounded semi-annually, then the sum of money is :

Let the Principal be Rs P

Simple interest = S.I. = $$\dfrac{P\times\ r\times\ t}{100}$$

or, S.I. = $$\dfrac{P\times\ 10\times\ \dfrac{3}{2}}{100}$$ = $$0.15 P$$

(time = $$1\dfrac{1}{2}$$ years = $$\dfrac{3}{2}$$ years)

Now calculating compound interest,

Principal = P

Rate = 10% p.a.

So, when compounded semi annually, rate = $$\left(\dfrac{10}{2}\%\right)$$= $$5\%$$

Also, in duration of $$1\dfrac{1}{2}$$ years, semi annually compounding will occur thrice.

So, compound interest = C.I. = $$P\left(1+\dfrac{5}{100}\right)^3-P$$

Now, according to question,

C.I. - S.I. = 183

So, $$P\left(1+\dfrac{5}{100}\right)^3-P$$ - $$0.15 P$$ = 183

or, $$P\left(1.05\right)^3-P-0.15P=183$$

or, $$0.007625P=183$$

or, $$P=₹24000$$