The difference between simple interest and compound interest on a certain sum is Rs 54.40 for 2 years at 8 percent per annum. Find the sum.

Answer

To solve the problem of finding the principal sum when the difference between compound interest (CI) and simple interest (SI) for 2 years at an interest rate of 8% is Rs 54.40, we can follow these steps:

### Step 1: Understand the given information
We know:
- The difference between CI and SI for 2 years is Rs 54.40.
- The rate of interest (r) is 8% per annum.
- The time (t) is 2 years.

### Step 2: Write the formula for Simple Interest (SI)
The formula for Simple Interest is:
\[ SI = \frac{P \times r \times t}{100} \]
Where:
- \( P \) = Principal amount
- \( r \) = Rate of interest
- \( t \) = Time in years

### Step 3: Substitute the values into the SI formula
Let the principal amount be \( P = x \).
\[ SI = \frac{x \times 8 \times 2}{100} = \frac{16x}{100} = \frac{4x}{25} \]

### Step 4: Write the formula for Compound Interest (CI)
The formula for Compound Interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where \( A \) is the total amount after time \( t \).

### Step 5: Calculate the amount (A) for CI
Substituting the values into the formula:
\[ A = x \left(1 + \frac{8}{100}\right)^2 = x \left(1 + 0.08\right)^2 = x \left(1.08\right)^2 \]
Calculating \( (1.08)^2 \):
\[ (1.08)^2 = 1.1664 \]
Thus,
\[ A = x \times 1.1664 \]

### Step 6: Find the Compound Interest (CI)
The Compound Interest can be calculated as:
\[ CI = A - P = x \times 1.1664 - x = x(1.1664 - 1) = x \times 0.1664 \]

### Step 7: Set up the equation for the difference between CI and SI
According to the problem:
\[ CI - SI = 54.40 \]
Substituting the expressions for CI and SI:
\[ x \times 0.1664 - \frac{4x}{25} = 54.40 \]

### Step 8: Simplify the equation
First, convert \( \frac{4x}{25} \) to a decimal:
\[ \frac{4}{25} = 0.16 \]
So the equation becomes:
\[ 0.1664x - 0.16x = 54.40 \]
Calculating the left side:
\[ (0.1664 - 0.16)x = 54.40 \]
\[ 0.0064x = 54.40 \]

### Step 9: Solve for \( x \)
To find \( x \):
\[ x = \frac{54.40}{0.0064} \]
Calculating this gives:
\[ x = 8500 \]

### Conclusion
The principal sum is Rs 8500.

---