If a sum amounts to ₹2,190 in four years and ₹2,409 in five years at compound interest, when the interest is compounded yearly, then the annual rate of interest is:

Let the rate of interest is R% per annum.

Let the principle amount =P

$$A =P(1+\dfrac{R}{100})^4$$

$$A_1=P(1+\dfrac{R}{100})^4$$----(i)

Similarly,

$$A_2=P(1+\dfrac{R}{100})^5 $$----(ii)

From equation(i) and (ii)

$$\Rightarrow \dfrac{A_1}{A_2}=\dfrac{P(1+\dfrac{R}{100})^4}{P(1+\dfrac{R}{100})^5}$$

$$\Rightarrow \dfrac{A_1}{A_2}=\dfrac{1}{(1+\dfrac{R}{100})}$$

$$\Rightarrow \dfrac{A_1}{A_2}=\dfrac{1}{(1+\dfrac{R}{100})}$$

$$\Rightarrow \dfrac{A_2}{A_1}=(1+\dfrac{R}{100})$$

$$\Rightarrow \dfrac{A_2}{A_1}-1=\dfrac{R}{100}$$

$$\Rightarrow \dfrac{A_2-A_1}{A_1}=\dfrac{R}{100}$$

$$\Rightarrow \dfrac{2409-2190}{2190}=\dfrac{R}{100}$$

$$\Rightarrow \dfrac{21900}{2190}=R$$

$$\Rightarrow R=10\%$$