The compound interest compounded annually, on a certain sum is ₹ 29,040 in second year and is ₹ 31,944 in third year.Calculate:

(i) the rate of interest.

(ii) the interest for 4th year.

(iii) the interest for 1st year.

(i) The difference between the amounts of two successive years = ₹ 31,944 - ₹ 29,040 = ₹ 2,904

⇒ ₹ 2,904 is the interest of one year on ₹ 29040.

P = ₹ 29,040, R = R %, T = 1 year, I = ₹ 2,904

$\text{Rate of interest} = \dfrac{100\times \text{I}}{\text{P}\times\text{T}}\\[1em] \Rightarrow R = \dfrac{100\times 2904}{29040 \times 1}\\[1em] \Rightarrow R = \dfrac{2,90400}{29,040}\\[1em] \Rightarrow R = 10$

Hence, the rate of interest = 10%.

(ii) Let P be the original principal.

For the first year:

P = ₹ P, R = 10 %, T = 1 year

$\text{Interest} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{P \times 10 \times 1}{100}\\[1em] = \dfrac{10P}{100}\\[1em] = \dfrac{P}{10}$

Amount at the end of first year = P + I

= ₹ P + $\dfrac{P}{10}$

= ₹ $\dfrac{11P}{10}$

For the second year:

P = ₹ $\dfrac{11P}{10}$, R = 10 %, T = 1 year, I = ₹ 29,040

$\text{Interest} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{\dfrac{11P}{10} \times 10 \times 1}{100}\\[1em] = \dfrac{11P}{100}$

29,040 = ₹ $\dfrac{11P}{100}$

⇒ P = ₹ $\dfrac{100 \times 29,040}{11} = \dfrac{29,04,000}{11}$ = ₹ 2,64,000

Principal amount for second year = ₹ $\dfrac{11}{10} \times 2,64,000$ = ₹ 2,90,400

Principal amount for third year = ₹ 2,90,400 + C.I. for second year = ₹ 2,90,400 + 29,040 = ₹ 3,19,440

Principal amount for fourth year = ₹ 55,770 + C.I. for third year = ₹ 3,19,440 + 31,944 = ₹ 3,51,384

For the fourth year:

P = ₹ 3,51,384, R = 10 %, T = 1 year

$\text{Interest} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{3,51,384 \times 10 \times 1}{100}\\[1em] = \dfrac{35,13,840}{100}\\[1em] = 35,138.4$

Hence, the interest at the end of fourth year = ₹ 35,138.4.

(iii) For the first year:

P = ₹ 2,64,000, R = 10 %, T = 1 year

$\text{Interest} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{₹ 2,64,000 \times 10 \times 1}{100}\\[1em] = ₹ 26,400$

Hence, the interest for 1st year = ₹ 26,400.