Simple & Compound Interest (Solution)

Solution 1:
\[Let\;\;p > 0\;\;be\;the\;principal\;and\;\;t > 0\;\;be\;the\;loan\;period\;in\;years.\]
\begin{align}
Interest\;Rate & = 6\%\;per\;annum \\
& = \frac{6}{12}\%\;per\;month \\
& = 0.5\%\;per\;month \\
\end{align}
\begin{align}
Total\;Amount\;(by\;Year) & = p \left( 1 + \frac{6t}{100} \right) \\
& = p \left( 1 + \frac{3t}{50} \right) \\
\end{align}
\begin{align}
Total\;Amount\;(by\;Month) & = p \left( 1 + \frac{0.5 \times 12t}{100} \right) \\
& = p \left( 1 + \frac{6t}{100} \right) \\
& = p \left( 1 + \frac{3t}{50} \right) \\
\end{align}
\begin{align}
Since\;Total\;Amount\;(by\;Year) & = Total\;Amount\;(by\;Month), \\
then\;Interest\;(by\;Year) & = Interest\;(by\;Month) \\
\end{align}


Solution 2:
\[Let\;\;p > 0\;\;be\;the\;principal\;and\;\;t > 0\;\;be\;the\;loan\;period\;in\;years.\]
\begin{align}
Interest\;Rate & = 6\%\;per\;annum \\
& = \frac{6}{12}\%\;per\;month \\
& = 0.5\%\;per\;month \\
\end{align}
\begin{align}
Total\;Amount\;(by\;Year) & = p \left( 1 + \frac{6}{100} \right)^t \\
& = p \left( \frac{53}{50} \right)^t \\
\end{align}
\begin{align}
Total\;Amount\;(by\;Month) & = p \left( 1 + \frac{0.5}{100} \right)^{12t} \\
& = p \left( 1 + \frac{1}{200} \right)^{12t} \\
& = p \left( \frac{201}{200} \right)^{12t} \\
\end{align}
\begin{align}
\frac{p \left( \frac{53}{50} \right)^t}{p \left( \frac{201}{200} \right)^{12t}} & = {\left( {\frac{{\frac{{53}}{{50}}}}{{{{\left( {\frac{{201}}{{200}}} \right)}^{12}}}}} \right)^t} \\
& = {\left( {\frac{{53{{\left( {200} \right)}^{12}}}}{{50{{\left( {201} \right)}^{12}}}}} \right)^t} \\
& = {\left( {\frac{{2.171 \times {{10}^{29}}}}{{2.174 \times {{10}^{29}}}}} \right)^t} \\
\end{align}
\begin{align}
Since\;{\left( {\frac{{2.171 \times {{10}^{29}}}}{{2.174 \times {{10}^{29}}}}} \right)^t} < 1,\;then & \frac{p \left( \frac{53}{50} \right)^t}{p \left( \frac{201}{200} \right)^{12t}} < 1 \\ \Rightarrow & p \left( \frac{53}{50} \right)^t < p \left( \frac{201}{200} \right)^{12t} \\ \Rightarrow & Total\;Amount\;(by\;Year) < Total\;Amount\;(by\;Month) \\ \Rightarrow & Interest\;(by\;Year) < Interest\;(by\;Month) \end{align}