The difference between compound interest and simple interest on Rs X at 6.5% per annum for 2 years is Rs 33.80. What is the value of X?

Options

A. Rs 7800
B. Rs 7500
C. Rs 8000
D. Rs 8500

Answer

To find the value of \( X \) such that the difference between compound interest (CI) and simple interest (SI) on \( Rs. X \) at a rate of \( 6.5\% \) per annum for \( 2 \) years is \( Rs. 33.80 \), we can follow these steps:

### Step 1: Understand the formula for Simple Interest (SI) and Compound Interest (CI)

1. **Simple Interest (SI)** for \( 2 \) years 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.

2. **Compound Interest (CI)** for \( 2 \) years can be calculated using the formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
This can be simplified to:
\[ CI = P \left( \left(1 + \frac{R}{100}\right)^T - 1 \right) \]

### Step 2: Calculate SI and CI for \( 2 \) years

- For \( 2 \) years, the SI can be calculated as:
\[ SI = \frac{X \times 6.5 \times 2}{100} = \frac{13X}{100} \]

- For \( 2 \) years, the CI can be calculated as:
\[ CI = X \left( \left(1 + \frac{6.5}{100}\right)^2 - 1 \right) \]
First, calculate \( \left(1 + \frac{6.5}{100}\right)^2 \):
\[ = \left(1 + 0.065\right)^2 = (1.065)^2 = 1.127225 \]
Therefore,
\[ CI = X \left(1.127225 - 1\right) = X \times 0.127225 \]

### Step 3: Find the difference between CI and SI

The difference between CI and SI is given as:
\[ CI - SI = 33.80 \]
Substituting the values we calculated:
\[ X \times 0.127225 - \frac{13X}{100} = 33.80 \]

### Step 4: Solve the equation

Rearranging the equation:
\[ X \times 0.127225 - 0.13X = 33.80 \]
This simplifies to:
\[ X \times (0.127225 - 0.13) = 33.80 \]
Calculating \( 0.127225 - 0.13 \):
\[ 0.127225 - 0.13 = -0.002775 \]
Thus, we have:
\[ -0.002775X = 33.80 \]
Dividing both sides by \(-0.002775\):
\[ X = \frac{33.80}{-0.002775} \approx 12100 \]
However, this doesn't seem correct. Let's check the calculations again.

### Step 5: Correct calculation of the difference

The correct difference should be calculated as:
\[ CI - SI = 33.80 \]
Substituting:
\[ X \times 0.127225 - \frac{13X}{100} = 33.80 \]
This gives:
\[ X(0.127225 - 0.13) = 33.80 \]
Calculating:
\[ 0.127225 - 0.13 = -0.002775 \]
Thus:
\[ X = \frac{33.80}{0.002775} \approx 12100 \]

### Final Calculation

To find the correct value of \( X \):
\[ 33.80 = X \times \left( \frac{6.5^2}{10000} \right) \]
Calculating \( \frac{6.5^2}{10000} \):
\[ = \frac{42.25}{10000} = 0.004225 \]
Thus:
\[ X = \frac{33.80}{0.004225} \approx 8000 \]

### Conclusion

The value of \( X \) is \( Rs. 8000 \).