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# Find the minimum number of complete years in which a sum of money put at $20\%$ compound interest will be more than doubled.

Hint: Assume in n complete years a sum of money (say P) put at $20\%$ compound interest will be more than doubled. Then with the help of the compound interest formula find the total amount after n years. According to the question, this amount will be twice the principal. Form an equation and solve for the value of n.

If principal=P , rate of compound interest=$R\%$ per annum, time =n years, then
Amount $= P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$
Now let us assume, in n complete years, a sum of money (say P) put at $20\%$ compound interest will be more than doubled.
$P{\left( {1 + \dfrac{{20}}{{100}}} \right)^n} > 2P \\ {\text{On simplifying we get,}} \\ \Rightarrow {\left( {\dfrac{6}{5}} \right)^n} > 2 \\ {\text{Taking log on both sides,we get}} \\ n\log \left( {\dfrac{6}{5}} \right) > \log 2 \\ {\text{As log}}\left( {\dfrac{6}{5}} \right) = 0.079,{\text{ we get,}} \\ \Rightarrow {\text{n x }}0.079 > 0.301 \\ \Rightarrow {\text{n > 3}}{\text{.81}} \\ \Rightarrow {\text{n}} \sim {\text{4}} \\$
Therefore the minimum number of complete years in which a sum of money put at $20\%$ compound interest will be more than doubled, is 4 years.
1.Amount= $P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$, when interest is compounded annually
2.Amount= $P{\left( {1 + \dfrac{{\dfrac{R}{2}}}{{100}}} \right)^{2n}}$, when interest is compounded half-yearly
3.Amount= $P{\left( {1 + \dfrac{{\dfrac{R}{4}}}{{100}}} \right)^{4n}}$, when interest is compounded quarterly