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