The compound interest is ? 6.40 more than the simple interest, if a sum is lent for 2 yr at 8% compound interest. Find the sum

Options

A. ? 1800
B. ? 10000
C. ? 800
D. ? 1000

Answer

To solve the problem step by step, we need to find the sum (P) that was lent out, given that the compound interest (CI) is ₹6.40 more than the simple interest (SI) for a period of 2 years at an interest rate of 8%.

### Step 1: 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 (the sum lent)
- \( R \) = rate of interest (8%)
- \( T \) = time (2 years)

### Step 2: Calculate Simple Interest (SI)
Substituting the values into the formula:
\[ SI = \frac{P \times 8 \times 2}{100} = \frac{16P}{100} = \frac{4P}{25} \]

### Step 3: Write the formula for Compound Interest (CI)
The formula for Compound Interest is:
\[ CI = A - P \]
where \( A \) is the amount after 2 years, calculated as:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = P \left(1 + \frac{8}{100}\right)^2 = P \left(1.08\right)^2 \]
Calculating \( (1.08)^2 \):
\[ (1.08)^2 = 1.1664 \]
Thus,
\[ A = 1.1664P \]
Now substituting back into the CI formula:
\[ CI = 1.1664P - P = 0.1664P \]

### Step 4: Set up the equation based on the given information
According to the problem, the compound interest is ₹6.40 more than the simple interest:
\[ CI = SI + 6.40 \]
Substituting the expressions for CI and SI:
\[ 0.1664P = \frac{4P}{25} + 6.40 \]

### Step 5: Convert the fraction to a common denominator
To solve the equation, we need a common denominator. The common denominator for 25 and 1 is 25:
\[ 0.1664P = \frac{4P}{25} + 6.40 \]
Convert \( 0.1664P \):
\[ 0.1664P = \frac{1664P}{10000} \quad \text{(since } 0.1664 = \frac{1664}{10000}\text{)} \]
Now, we can rewrite the equation:
\[ \frac{1664P}{10000} = \frac{4P}{25} + 6.40 \]

### Step 6: Clear the fractions
Multiply the entire equation by 10000 to eliminate the denominators:
\[ 1664P = 400P + 64000 \]

### Step 7: Solve for P
Rearranging gives:
\[ 1664P - 400P = 64000 \]
\[ 1264P = 64000 \]
Now divide both sides by 1264:
\[ P = \frac{64000}{1264} \approx 50.63 \]

### Step 8: Round to the nearest whole number
Since we typically deal with whole amounts in currency, we round:
\[ P \approx 50 \]

### Conclusion
The sum lent out is approximately ₹50.