A sum of money becomes eight times in 3 years, If the rate is compounded annually. In how much time will the same amount at the same compound rate become sixteen times ?

$$\eqalign{ & {\text{Let principal = P}} \cr & {\bf{Case (I)}} \cr & {\text{Time = 3 years,}} \cr & {\text{Amount = 8P}} \cr & \Rightarrow 8{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & \Rightarrow {\left( 2 \right)^3} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & {\text{Taking cube root of both sides,}} \cr & \Rightarrow {\text{2 = }}\left( {1 + \frac{{\text{R}}}{{100}}} \right) \cr & \Rightarrow {\text{R = 100 }}\% \cr & {\bf{Case (II)}} \cr & {\text{Let after t years it will be 16 times}} \cr & \Rightarrow 16{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{\text{t}}} \cr & \Rightarrow 16 = {\left( 2 \right)^{\text{t}}} \cr & \Rightarrow {\left( 2 \right)^4} = {\left( 2 \right)^{\text{t}}} \cr & \Rightarrow {\text{t}} = 4 \cr & {\text{Hence required time}} \cr & {\text{(t) = 4 years}} \cr} $$