Question

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

Answer 1

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.

Complete step-by-step answer:
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.
Therefore,
\[
  P{\left( {1 + \dfrac{{20}}{{100}}} \right)^n} > 2P \\
  {\text{On simplifying we get,}} \\
   \Rightarrow {\left( {\dfrac{6}{5}} \right)^n} > 2 \\
  {\text{Taking log on both sides,we get}} \\
  n\log \left( {\dfrac{6}{5}} \right) > \log 2 \\
  {\text{As log}}\left( {\dfrac{6}{5}} \right) = 0.079,{\text{ we get,}} \\
   \Rightarrow {\text{n x }}0.079 > 0.301 \\
   \Rightarrow {\text{n > 3}}{\text{.81}} \\
   \Rightarrow {\text{n}} \sim {\text{4}} \\
  \]
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.

Note: Some important formulae to remember:
If principal=P , rate of compound interest=R% per annum, time =n years, then
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