If a certain sum becomes 3 times in 6 years at compound interest, then in how many years, it will become 81 times?

Let principal sum = Rs. $$P$$ and rate of interest = $$r\%$$

Amount under compound interest = $$P(1+\frac{r}{100})^T$$

Thus, after 6 years

=> $$P(1+\frac{r}{100})^6=3P$$

=> $$(1+\frac{r}{100})^6=3$$

=> $$(1+\frac{r}{100})=(3)^{\frac{1}{6}}$$ -------------(i)

Now, Let after $$t$$ years sum becomes 81 times

=> $$P(1+\frac{r}{100})^t=81P$$

=> $$(3)^{\frac{t}{6}}=(3)^4$$

Comparing the exponents, we get :

=> $$\frac{t}{6}=4$$

=> $$t=4\times6=24$$ years

=> Ans - (D)