On what sum of money will the difference between simple intersect and compound intersect for 2 years at 5% per annum be equal to Rs 25 ?

Answer

To solve the problem step by step, we will find the principal amount (P) for which the difference between compound interest (CI) and simple interest (SI) for 2 years at a rate of 5% per annum equals Rs 25.

### Step 1: Write down the formulas for Simple Interest and Compound Interest
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
- The formula for Compound Interest (CI) is:
\[ CI = A - P \]
where \( A \) (the amount) is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

### Step 2: Substitute the known values into the Simple Interest formula
Given:
- Rate (R) = 5%
- Time (T) = 2 years

Substituting these values into the SI formula:
\[ SI = \frac{P \times 5 \times 2}{100} = \frac{10P}{100} = \frac{P}{10} \]

### Step 3: Substitute the known values into the Compound Interest formula
Using the formula for \( A \):
\[ A = P \left(1 + \frac{5}{100}\right)^2 = P \left(1 + 0.05\right)^2 = P \left(1.05\right)^2 \]
Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]
Thus,
\[ A = P \times 1.1025 \]
Now substituting into the CI formula:
\[ CI = A - P = P \times 1.1025 - P = P(1.1025 - 1) = P \times 0.1025 \]

### Step 4: Find the difference between CI and SI
We know that:
\[ CI - SI = 25 \]
Substituting the expressions for CI and SI:
\[ P \times 0.1025 - \frac{P}{10} = 25 \]

### Step 5: Simplify the equation
Convert \(\frac{P}{10}\) to a decimal:
\[ \frac{P}{10} = 0.1P \]
Now the equation becomes:
\[ 0.1025P - 0.1P = 25 \]
Simplifying further:
\[ (0.1025 - 0.1)P = 25 \]
\[ 0.0025P = 25 \]

### Step 6: Solve for P
To find \( P \):
\[ P = \frac{25}{0.0025} = 10000 \]

### Final Answer
The principal amount (P) is Rs 10,000.

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