The difference between C.I. and S.I. on a certain sum of money at 10% per annum for 3 years is 620. Find the principal if it is known that the interest is compounded annually.

Options

A. 200000
B. 20000
C. 10000
D. 100000

Answer

To solve the problem step by step, we will use the formula for the difference between Compound Interest (C.I.) and Simple Interest (S.I.) over a certain period.

### Step 1: Understand the given information
We know:
- The difference between C.I. and S.I. is 620.
- The rate of interest (R) is 10% per annum.
- The time (T) is 3 years.

### Step 2: Write the formula for the difference between C.I. and S.I.
The formula for the difference between C.I. and S.I. is given by:
\[ C.I. - S.I. = \frac{P \times R^2 \times T(T + 1)}{100^2} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

### Step 3: Substitute the known values into the formula
We substitute the known values into the formula:
\[ 620 = \frac{P \times (10)^2 \times 3(3 + 1)}{100^2} \]
This simplifies to:
\[ 620 = \frac{P \times 100 \times 12}{10000} \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 620 = \frac{P \times 1200}{10000} \]
\[ 620 = \frac{P \times 12}{100} \]

### Step 5: Cross-multiply to solve for P
Cross-multiplying gives:
\[ 620 \times 100 = P \times 12 \]
\[ 62000 = 12P \]

### Step 6: Solve for P
Now, divide both sides by 12 to find \( P \):
\[ P = \frac{62000}{12} \]
Calculating this gives:
\[ P = 5166.67 \]

### Step 7: Finalize the answer
Thus, the principal amount \( P \) is approximately 5166.67.