If the compound interest on a certain sum for 2 years at the rate of 10% is Rs. 105 and simple interest is Rs. 100, then the sum is

Answer

To solve the problem step by step, we will find the principal amount (sum) based on the given compound interest (CI) and simple interest (SI).

### Step 1: Understand the given information
We know:
- Compound Interest (CI) for 2 years = Rs. 105
- Simple Interest (SI) for 2 years = Rs. 100
- Rate of interest (R) = 10%
- Time (T) = 2 years

### Step 2: Calculate the difference between CI and SI
The difference between the compound interest and simple interest over 2 years can be calculated as follows:
\[ \text{Difference} = \text{CI} - \text{SI} = 105 - 100 = Rs. 5 \]

### Step 3: Use the formula for the difference between CI and SI
The difference between compound interest and simple interest for 2 years can be expressed using the formula:
\[ \text{Difference} = \frac{P \times R^2}{100^2} \]
Where \( P \) is the principal amount and \( R \) is the rate of interest.

### Step 4: Substitute the known values into the formula
We know the difference is Rs. 5, and the rate of interest is 10%. Substituting these values into the formula gives:
\[ 5 = \frac{P \times (10)^2}{100^2} \]
This simplifies to:
\[ 5 = \frac{P \times 100}{10000} \]

### Step 5: Simplify the equation
This can be further simplified:
\[ 5 = \frac{P}{100} \]

### Step 6: Solve for the principal amount (P)
To find \( P \), we can rearrange the equation:
\[ P = 5 \times 100 = Rs. 500 \]

### Conclusion
The sum (principal amount) is Rs. 500.

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