The difference between the compound interest and the simple interest on a certain sum for 2 years at 6% per annum is Rs 90. Find the sum.

Answer

To solve the problem step by step, we will find the sum (principal amount) using the given information about the difference between compound interest (CI) and simple interest (SI) over 2 years at a rate of 6% per annum.

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

### Step 2: Calculate Simple Interest for 2 years
Given:
- Rate \( R = 6\% \)
- Time \( T = 2 \) years

Using the formula for SI:
\[ SI = \frac{P \times 6 \times 2}{100} = \frac{12P}{100} = \frac{3P}{25} \]

### Step 3: Calculate Compound Interest for 2 years
Using the formula for A:
\[ A = P \left(1 + \frac{6}{100}\right)^2 = P \left(1 + 0.06\right)^2 = P \left(1.06\right)^2 \]
Calculating \( (1.06)^2 \):
\[ (1.06)^2 = 1.1236 \]
Thus,
\[ A = 1.1236P \]
Now, calculate CI:
\[ CI = A - P = 1.1236P - P = 0.1236P \]

### Step 4: Find the difference between CI and SI
The difference between CI and SI is given as Rs 90:
\[ CI - SI = 90 \]
Substituting the values we found:
\[ 0.1236P - \frac{3P}{25} = 90 \]

### Step 5: Find a common denominator and solve for P
Convert \( \frac{3P}{25} \) to a decimal:
\[ \frac{3P}{25} = 0.12P \]
Now, substituting back:
\[ 0.1236P - 0.12P = 90 \]
Calculating the left side:
\[ (0.1236 - 0.12)P = 90 \]
\[ 0.0036P = 90 \]

### Step 6: Solve for P
Now, divide both sides by 0.0036:
\[ P = \frac{90}{0.0036} = 25000 \]

### Conclusion
The principal amount (sum) is Rs 25,000.

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