The difference berween the compount interest, compounded annually and the simple interest on a certain sum for 2 years at 15% per annum is Rs 180. Find the sum.

Answer

To solve the problem step by step, we will find the principal amount (sum) for which the difference between compound interest (CI) and simple interest (SI) for 2 years at 15% per annum is Rs 180.

### Step 1: Understand the formulas
- **Simple Interest (SI)** is calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

- **Compound Interest (CI)** for 2 years can be calculated using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where \( A \) is the amount after \( T \) years. The compound interest can then be found as:
\[ CI = A - P \]

### Step 2: Calculate Simple Interest for 2 years
Given:
- Rate \( R = 15\% \)
- Time \( T = 2 \) years

Using the SI formula:
\[ SI = \frac{P \times 15 \times 2}{100} = \frac{30P}{100} = \frac{3P}{10} \]

### Step 3: Calculate Compound Interest for 2 years
Using the CI formula:
\[ A = P \left(1 + \frac{15}{100}\right)^2 = P \left(1.15\right)^2 \]
Calculating \( (1.15)^2 \):
\[ (1.15)^2 = 1.3225 \]
Thus,
\[ A = P \times 1.3225 \]
Now, calculating CI:
\[ CI = A - P = P \times 1.3225 - P = P(1.3225 - 1) = P \times 0.3225 \]

### Step 4: Set up the equation for the difference
We know from the problem statement that:
\[ CI - SI = 180 \]
Substituting the values we calculated:
\[ P \times 0.3225 - \frac{3P}{10} = 180 \]

### Step 5: Convert \(\frac{3P}{10}\) to a decimal
\[ \frac{3P}{10} = 0.3P \]
So the equation becomes:
\[ P \times 0.3225 - P \times 0.3 = 180 \]
This simplifies to:
\[ P \times (0.3225 - 0.3) = 180 \]
Calculating \( 0.3225 - 0.3 \):
\[ 0.3225 - 0.3 = 0.0225 \]
Thus, we have:
\[ P \times 0.0225 = 180 \]

### Step 6: Solve for \( P \)
To find \( P \):
\[ P = \frac{180}{0.0225} \]
Calculating this gives:
\[ P = 8000 \]

### Final Answer
The principal amount (sum) is Rs 8000.

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