The difference between the simple interest and the compound interest earned on a sum of money at the end of four years at the rate of `10%` p.a. is Rs. `256cdot40`. What is the sum ?

Options

A. Rs 4000
B. Rs. 4500
C. Rs. 5000
D. Rs. 6000

Answer

To find the sum of money (P) based on the difference between simple interest (SI) and compound interest (CI) over 4 years at a rate of 10% per annum, we can follow these steps:

### Step 1: Understand the relationship between SI and CI
The difference between the compound interest (CI) and simple interest (SI) for a principal amount (P) over time (T) at a rate (R) can be given by the formula:
\[ \text{Difference} = CI - SI = P \left( \frac{R}{100} \right)^2 \cdot \frac{T(T-1)}{2} \]
For 4 years at 10%:
\[ \text{Difference} = P \left( \frac{10}{100} \right)^2 \cdot \frac{4(4-1)}{2} \]

### Step 2: Substitute the values
Substituting the values into the formula:
\[ \text{Difference} = P \left( \frac{10}{100} \right)^2 \cdot \frac{4 \cdot 3}{2} \]
Calculating the values:
\[ = P \cdot \left( \frac{1}{10} \right)^2 \cdot 6 = P \cdot \frac{1}{100} \cdot 6 = \frac{6P}{100} = \frac{3P}{50} \]

### Step 3: Set the equation equal to the given difference
We know from the problem that the difference is Rs. 256.40:
\[ \frac{3P}{50} = 256.40 \]

### Step 4: Solve for P
To find P, we can multiply both sides by 50:
\[ 3P = 256.40 \times 50 \]
Calculating the right side:
\[ 3P = 12820 \]
Now, divide both sides by 3:
\[ P = \frac{12820}{3} = 4273.33 \]

### Step 5: Final answer
The sum of money (P) is approximately Rs. 4273.33.