The difference between the interest earned under compound interest, interest being compounded annually and simple interest for two years on the same sum and at the same rate of interest is Rs. \[25.60\]. Find the sum if the rate of interest is \[8\% \] p.a. (in Rs.).
A.2000
B.2500
C.3200
D.4000

Hint: Here, we have to find the sum or the principal amount. First, we will find the simple interest and compound interest using the respective formula. We will then subtract the simple interest from the compound and equate it to the given difference between these interests to form an equation. We will solve this equation further to get the required sum.

Formula Used:
We will use the following formulas :
1.Compound interest being compounded annually, then amount is given by the formula \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}\] where \[P\] is the principal, \[r\] is the rate of interest and \[t\] is the number of years.
2.Compound interest being compounded annually, then interest is given by the formula \[C.I. = A - P\] where \[P\] is the principal, \[A\] is the amount.
3.Simple Interest is given by the formula \[S.I. = \dfrac{{P \times r \times t}}{{100}}\] , where \[P\] is the principal, \[r\] is the rate of interest and \[t\] is the number of years.

Complete step-by-step answer:
We are given that the Rate of Interest \[r = 8\% \] and number of years, \[t = 2\] .
Let the principal be \[P\].
Now, we have to find the simple interest for the principal with the given rate of interest and number of years.
Substituting \[r = 8\% \] and \[t = 2\] in the formula \[S.I. = \dfrac{{P \times r \times t}}{{100}}\], we get
\[S.I. = \dfrac{{P \times 8 \times 2}}{{100}}\]
Multiplying the terms in the numerator, we get
\[ \Rightarrow S.I. = \dfrac{{P \times 16}}{{100}}\]
Dividing 16 by 100, we get
\[ \Rightarrow S.I. = 0.16P\]
Now, we have to find the amount with the given rate of interest and number of years.
Substituting \[r = 8\% \] and \[t = 2\] in the formula\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}\], we get
\[A = P{\left( {1 + \dfrac{8}{{100}}} \right)^2}\]
On cross-multiplying, we get
\[ \Rightarrow A = P{\left( {1 \times \dfrac{{100}}{{100}} + \dfrac{8}{{100}}} \right)^2}\]
Adding the terms, we get
\[ \Rightarrow A = P{\left( {\dfrac{{108}}{{100}}} \right)^2}\]
Squaring the terms, we get
\[ \Rightarrow A = P\left( {\dfrac{{11664}}{{10000}}} \right)\]
Dividing 11664 by 10000, we get
\[ \Rightarrow A = P(1.1664)\]
Now we have to find the compound interest, so substituting \[A = P(1.1664)\] in the formula\[C.I. = A - P\], we get
\[C.I. = (1.1664)P - P\]
\[ \Rightarrow C.I. = (0.1664)P\]
Since the difference between the interest earned under compound interest, interest being compounded annually and simple interest for two years on the same sum and at the same rate of interest is Rs. \[25.60\]. So,
\[C.I. - S.I. = 25.60\]
Substituting \[C.I. = 0.1664P\] and \[S.I. = 0.16P\] in the above equation, we get
\[ \Rightarrow 0.1664P - 0.16P = 25.60\]
Subtracting the terms, we get
\[ \Rightarrow 0.0064P = 25.60\]
Multiplying by 10000 on both the sides, we get
\[ \Rightarrow 64P = 256000\]
Dividing by 64 on both the sides, we get
\[ \Rightarrow P = \dfrac{{256000}}{{64}}\]
\[ \Rightarrow P = 4000\]
Therefore, the sum is Rs. 4000.
Hence, option D is the correct answer.

Note: To solve this question, we need to know the difference between both simple interest and compound interest. The major difference between simple interest and compound interest is that simple interest is calculated on principal amount only whereas compound interest is calculated on the principal amount as well as the interest compounded for a cycle of a period. The interest charged on simple interest is for the principal amount whereas the interest charged on compound interest is for the principal and accumulated interest.