The difference between compound interest and simple interested on a sum of money for two years is given by

Options

A. `(Pr^(2))/(100)`
B. `(Pr^(2))/(50)`
C. `(2Pr)/(10000)`
D. `(Pr^(2))/(10000)`

Answer

To find the difference between compound interest (CI) and simple interest (SI) on a sum of money for two years, we can follow these steps:

### Step 1: Define the Variables
Let:
- Principal amount = \( P \)
- Rate of interest = \( r \% \)
- Time = 2 years

### Step 2: Calculate Simple Interest for 2 Years
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
For 2 years, this becomes:
\[ \text{SI} = \frac{P \times r \times 2}{100} = \frac{2Pr}{100} \]

### Step 3: Calculate Compound Interest for 2 Years
The formula for Compound Interest (CI) is:
\[ \text{CI} = A - P \]
where \( A \) is the amount after 2 years, given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
For 2 years, this becomes:
\[ A = P \left(1 + \frac{r}{100}\right)^2 \]
Expanding this using the binomial theorem:
\[ A = P \left(1 + \frac{r}{100}\right)^2 = P \left(1 + 2 \cdot \frac{r}{100} + \left(\frac{r}{100}\right)^2\right) = P \left(1 + \frac{2r}{100} + \frac{r^2}{10000}\right) \]
Thus,
\[ A = P + \frac{2Pr}{100} + \frac{Pr^2}{10000} \]
Now, substituting back into the CI formula:
\[ \text{CI} = A - P = \left(P + \frac{2Pr}{100} + \frac{Pr^2}{10000}\right) - P = \frac{2Pr}{100} + \frac{Pr^2}{10000} \]

### Step 4: Find the Difference Between CI and SI
Now, we can find the difference:
\[ \text{Difference} = \text{CI} - \text{SI} \]
Substituting the values we calculated:
\[ \text{Difference} = \left(\frac{2Pr}{100} + \frac{Pr^2}{10000}\right) - \frac{2Pr}{100} \]
The \( \frac{2Pr}{100} \) terms cancel out:
\[ \text{Difference} = \frac{Pr^2}{10000} \]

### Final Result
Thus, the difference between compound interest and simple interest for two years is:
\[ \text{Difference} = \frac{Pr^2}{10000} \]