Find the sum on which the difference between the simple interest and the compound interest at the rate of 8% per annum compounded annually be रु 64 in 2 years .

Explanation

To find the sum on which the difference between the simple interest (SI) and the compound interest (CI) at the rate of 8% per annum compounded annually is ₹64 in 2 years, we can follow these steps:

### Step 1: Define the variables
Let the principal amount (sum) be denoted as \( P \).

### Step 2: Calculate Simple Interest (SI)
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R = 8\% \) (rate of interest)
- \( T = 2 \) years (time)

Substituting the values:
\[ SI = \frac{P \times 8 \times 2}{100} = \frac{16P}{100} = \frac{4P}{25} \]

### Step 3: Calculate Compound Interest (CI)
The formula for the amount \( A \) after \( n \) years with compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Substituting the values:
\[ A = P \left(1 + \frac{8}{100}\right)^2 = P \left(1 + 0.08\right)^2 = P \left(1.08\right)^2 \]
Calculating \( (1.08)^2 \):
\[ (1.08)^2 = 1.1664 \]
Thus, the amount \( A \) is:
\[ A = 1.1664P \]
Now, the Compound Interest (CI) can be calculated as:
\[ CI = A - P = 1.1664P - P = 0.1664P \]

### Step 4: Set up the equation for the difference between CI and SI
According to the problem, the difference between CI and SI is ₹64:
\[ CI - SI = 64 \]
Substituting the values we calculated:
\[ 0.1664P - \frac{4P}{25} = 64 \]

### Step 5: Convert \( \frac{4P}{25} \) to a decimal
To solve the equation, we need a common denominator. The decimal equivalent of \( \frac{4P}{25} \) is:
\[ \frac{4P}{25} = 0.16P \]
Now the equation becomes:
\[ 0.1664P - 0.16P = 64 \]

### Step 6: Simplify the equation
Subtracting the two terms:
\[ (0.1664 - 0.16)P = 64 \]
\[ 0.0064P = 64 \]

### Step 7: Solve for \( P \)
To find \( P \), divide both sides by \( 0.0064 \):
\[ P = \frac{64}{0.0064} \]
Calculating the right-hand side:
\[ P = 10000 \]

### Final Answer
The sum on which the difference between the simple interest and the compound interest is ₹64 in 2 years is **₹10,000**.

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