The difference between CI and SI on Rs.8000 for 2yr is Rs.20. Find the rate of interest.

Options

A. 0.05
B. 0.04
C. 0.02
D. 0.03

Answer

To find the rate of interest given that the difference between Compound Interest (CI) and Simple Interest (SI) on Rs.8000 for 2 years is Rs.20, we can follow these steps:

### Step 1: Understand the formulas for CI and SI
- Simple Interest (SI) for 2 years can be 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) for 2 years can be calculated using the formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

### Step 2: Calculate SI for 2 years
Given:
- Principal \( P = 8000 \)
- Time \( T = 2 \)

Using the SI formula:
\[ SI = \frac{8000 \times R \times 2}{100} = 160R \]

### Step 3: Calculate CI for 2 years
Using the CI formula:
\[ CI = 8000 \left(1 + \frac{R}{100}\right)^2 - 8000 \]
Expanding the expression:
\[ CI = 8000 \left(\left(1 + \frac{R}{100}\right)^2 - 1\right) \]
Using the binomial expansion:
\[ \left(1 + \frac{R}{100}\right)^2 = 1 + 2\frac{R}{100} + \left(\frac{R}{100}\right)^2 \]
Thus:
\[ CI = 8000 \left(2\frac{R}{100} + \left(\frac{R}{100}\right)^2\right) = 160R + \frac{8000R^2}{10000} = 160R + 0.8R^2 \]

### Step 4: Find the difference between CI and SI
We know that:
\[ CI - SI = 20 \]
Substituting the values we found:
\[ (160R + 0.8R^2) - 160R = 20 \]
This simplifies to:
\[ 0.8R^2 = 20 \]

### Step 5: Solve for R
Dividing both sides by 0.8:
\[ R^2 = \frac{20}{0.8} = 25 \]
Taking the square root:
\[ R = 5 \]

### Conclusion
The rate of interest \( R \) is \( 5\% \).