The difference between C.I. and S.I. on Rs.7500 for two years is Rs.12 at the same rate of interest per annum. Find the rate of interest.

Hint – To find the rate of interest, we take in all the given data in the question which are the principal amount, time period, difference between compound and simple interest and the rate of interest is the same for both of them. We substitute this data in the formulae of simple interest and compound interest to get the answer.

Complete step-by-step answer:
Given data,
Principle P = 7500/-
Rate of interest is same, let it be R
Difference between C.I and S.I = 12/-
Time = 2 years
We know simple interest is given by the formula,
${\text{S}}{\text{.I = }}\dfrac{{{\text{PRT}}}}{{100}}$, where P is the principle, R is the rate of interest and T is the time period.
Therefore the Simple interest in our case is,
${\text{S}}{\text{.I = }}\dfrac{{7500 \times {\text{R}} \times {\text{2}}}}{{100}} = 150{\text{R}}$
We know the compound interest is given by the formula,
${\text{C}}{\text{.I = P}}{\left( {1 + \dfrac{{\text{x}}}{{100}}} \right)^2} - {\text{P}}$, where P is the principle, x is the rate of interest.
Here according to the question rate of interest is R, i.e. x = R therefore the Compound interest in our case is,
${\text{C}}{\text{.I = P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^2} - {\text{P}}$
$
  {\text{C}}{\text{.I = 7500}}\left( {1 + \dfrac{{{{\text{R}}^2}}}{{10000}} + \dfrac{{2{\text{R}}}}{{100}} - 1} \right) \\
  {\text{C}}{\text{.I = }}\dfrac{{{\text{3}}{{\text{R}}^2}}}{4} + 150{\text{R}} \\
$

Given the difference between Compound Interest and Simple Interest = Rs.12
$
   \Rightarrow \dfrac{{{\text{3}}{{\text{R}}^2}}}{4} + 150{\text{R}} - 150{\text{R = 12}} \\
   \Rightarrow \dfrac{{{\text{3}}{{\text{R}}^2}}}{4} = 12 \\
   \Rightarrow {{\text{R}}^2} = 12 \times \dfrac{4}{3} \\
   \Rightarrow {\text{R = 4% }} \\
$
Hence the rate of interest is 4%.

Note – In order to solve this type of question the key is to know the definitions and formula of compound interest and simple interest. Here we were able to get both terms into a single variable because the rate of interest is the same.
Compound Interest is calculated by adding the interest to the principal amount after a specific time period. Therefore the principle keeps increasing over time.
Simple Interest is calculated by only counting interest on the actual amount irrespective of the duration of the loan taken. Principle is always constant.
An expansion of the form is given by: ${\left( {{\text{a + b}}} \right)^2} = {{\text{a}}^2}{\text{ + }}{{\text{b}}^2}{\text{ + 2ab}}$.