The difference between the simple interest and C.I. at the same rate of ${\rm{Rs}}{\rm{.}}\;{\rm{5000}}$ for 2 years is ${\rm{Rs}}{\rm{.}}\;{\rm{72}}$ . The rate of interest is
A. $10\% $
B.$12\% $
C.$6\% $
D.$8\% $

Hint: To find the rate of interest, we will find the expression for the difference between compound interest and simple interest. We will use the expression of simple interest to find the simple interest of 2 years. Since, we already know the formula to find the amount by the compound interest, we will subtract principal from this amount to find the compound interest for 2 years. Now we subtract compound interest and simple interest, then we will substitute the respective values given in the question to find the rate of interest.

Formula Used:
We already know the formula for simple interest, which is expressed as:
$S.I. = \dfrac{{P \times r \times t}}{{100}}$

Where, $P$ stands for principal, $r$ stands for rate of interest per annum and $t$ is the time in years.


Complete step by step solution
Given:

The principal amount is $P = {\rm{Rs}}{\rm{.}}\;{\rm{5000}}$.
We will assume $r\% $ as the rate of interest per annum and $t$ as time in years. We know the formula to find the simple interest is expressed as:
$S.I. = \dfrac{{P \times r \times t}}{{100}}$

For 1 year the simple interest can be written as:
\[S.I. = \dfrac{{\Pr }}{{100}}\]

For the second year simple interest can be written as
\[S.I. = \dfrac{{\Pr }}{{100}}\]

Now, the sum of simple interest for the 2 years is written as:
\[S.I. = 2P\left( {\dfrac{r}{{100}}} \right)\]……(i)

We know that the amount that we get by compound interest can be expressed as:
${\rm{Amount}} = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}$
Where, $n$ is the number of years.

To find the compound interest for 2 years we will subtract $P$ from the amount and substitute 2 for $n$, which can be expressed as:
$\begin{array}{l}
C.I. = {\rm{Amount}} - P\\
C.I. = P{\left( {1 + \dfrac{r}{{100}}} \right)^2} - P
\end{array}$

We will use binomial distribution to simplify the above expression.
$\begin{array}{l}
C.I. = P\left\{ {1 + 2\left( {\dfrac{r}{{100}}} \right) + {{\left( {\dfrac{r}{{100}}} \right)}^2}} \right\} - P\\
C.I. = P + 2P\left( {\dfrac{r}{{100}}} \right) + 2{\left( {\dfrac{r}{{100}}} \right)^2} - P\\
C.I. = 2P\left( {\dfrac{r}{{100}}} \right) + 2{\left( {\dfrac{r}{{100}}} \right)^2}
\end{array}$ ……(ii)

To find the difference between $C.I.$ and $S.I.$ we will subtract equation (i) from (ii), we get,
$\begin{array}{l}
C.I. - S.I. = 2P\left( {\dfrac{r}{{100}}} \right) + 2{\left( {\dfrac{r}{{100}}} \right)^2} - 2P\left( {\dfrac{r}{{100}}} \right)\\
C.I - S.I. = P{\left( {\dfrac{r}{{100}}} \right)^2}
\end{array}$

In the above expression, we will substitute \[{\rm{Rs}}{\rm{.}}\;{\rm{5000}}\] for $P$ and ${\rm{Rs}}{\rm{.}}\;{\rm{72}}$ for the difference between $C.I.$ and $S.I.$
${\rm{Rs}}{\rm{.}}\;72 = {\rm{Rs}}{\rm{.}}\;{\rm{5000}}{\left( {\dfrac{r}{{100}}} \right)^2}$

We rewrite the above expression as:
\[\begin{array}{c}
{\left( {\dfrac{r}{{100}}} \right)^2} = \dfrac{{{\rm{Rs}}{\rm{.}}\;72}}{{{\rm{Rs}}{\rm{.}}\;{\rm{5000}}}}\\
{r^2} = \dfrac{{{\rm{Rs}}{\rm{.}}\;72 \times 100 \times 100}}{{{\rm{Rs}}{\rm{.}}\;{\rm{5000}}}}\\
{r^2} = 144\\
r = \sqrt {144} \\
r = 12\% \;{\rm{per}}\;{\rm{annum}}
\end{array}\]

Hence, the correct option is B.


Note: To solve this question, we should have prior knowledge about the formulas of simple interest and compound interest. The formula we have for simple interest gives as directly simple interest but the formula that we have for compound interest gives us the amount that we will get from compound interest so in order to find the compound we need to subtract the principal from the amount.