The compound and simple interest on a certain sum is Rs. 209 and Rs. 200 respectively for 2 years. What is the rate percent?
A. 9%
B. 18%
C. $4.5$%
D. 10%

Hint: We first explain the formulas for simple and compound interest. We assume the values for the interest rate and the principal. We put the values and find the equations for two unknowns. We solve them to find the solutions.

Complete step by step answer:
First, we will explain the formulas for compound interest and simple interest.
Let the principal be P, interest rate be r and time period be n, then for the compound interest the formula will be $A=P{{\left( 1+\dfrac{r}{100} \right)}^{n}}-P$ and simple interest will be $A=\dfrac{Pnr}{100}$.
Let us assume the amount is P and the rate is r.
So, according to the formula the simple interest ${{A}_{1}}=\dfrac{P\times 2\times r}{100}=200$ and the compound interest ${{A}_{2}}=P{{\left( 1+\dfrac{r}{100} \right)}^{2}}-P=209$.
We now take $P$ common from the equation and use the identity formula of ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ to factor the given equation.
We take the representation of $a=1+\dfrac{r}{100};b=1$.
Simplifying ${{A}_{2}}=P{{\left( 1+\dfrac{r}{100} \right)}^{2}}-P=209$, we get
$\begin{align}
  & P\left( 1+\dfrac{r}{100}+1 \right)\left( 1+\dfrac{r}{100}-1 \right)=209 \\
 & \Rightarrow \dfrac{Pr}{100}\left( 2+\dfrac{r}{100} \right)=209 \\
\end{align}$
We know the other equation and try to find the part of the equation to replace in the main equation.
From ${{A}_{1}}=\dfrac{P\times 2\times r}{100}=200$, we get $Pr=10000$.
Putting the value in $\dfrac{Pr}{100}\left( 2+\dfrac{r}{100} \right)=209$, we get
$\begin{align}
  & \dfrac{Pr}{100}\left( 2+\dfrac{r}{100} \right)=209 \\
 & \Rightarrow 2+\dfrac{r}{100}=\dfrac{209}{100} \\
\end{align}$
We now take the variables and the constant on the separate sides to simplify the equation.
$\begin{align}
  & 2+\dfrac{r}{100}=\dfrac{209}{100} \\
 & \Rightarrow \dfrac{r}{100}=\dfrac{209}{100}-2=\dfrac{9}{100} \\
 & \Rightarrow r=9 \\
\end{align}$

So, the correct answer is “Option A”.

Note: We also can use the substitution process where we replace the values for one variable in the second equation. The addition of the interest with the principal value is the total payback. So, the formula becomes $P{{\left( 1+\dfrac{r}{100} \right)}^{n}}$ and $P+\dfrac{Pnr}{100}$.