The difference between CI and SI on ₹ 8000 for 2 yr is ₹ 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 ₹8000 for 2 years is ₹20, we can follow these steps:

### Step 1: Understand the formulas
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

- The formula for Compound Interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

### Step 2: Substitute the known values
Given:
- Principal \( P = 8000 \)
- Time \( T = 2 \) years
- Difference \( CI - SI = 20 \)

Using the formulas:
1. Calculate SI:
\[ SI = \frac{8000 \times R \times 2}{100} = \frac{16000R}{100} = 160R \]

2. Calculate CI:
\[ CI = 8000 \left(1 + \frac{R}{100}\right)^2 - 8000 \]

### Step 3: Expand the CI formula
Using the binomial expansion for \( (1 + x)^n \):
\[ (1 + \frac{R}{100})^2 = 1 + 2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2 \]
Thus,
\[ CI = 8000 \left(1 + 2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2\right) - 8000 \]
\[ = 8000 \left(2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2\right) \]
\[ = 8000 \cdot 2 \cdot \frac{R}{100} + 8000 \cdot \left(\frac{R}{100}\right)^2 \]
\[ = 160R + \frac{8000R^2}{10000} \]
\[ = 160R + \frac{8R^2}{10} \]
\[ = 160R + 0.8R^2 \]

### Step 4: Set up the equation
Now, we know that:
\[ CI - SI = 20 \]
Substituting the values we found:
\[ (160R + 0.8R^2) - 160R = 20 \]
\[ 0.8R^2 = 20 \]

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

### Step 6: Find the rate of interest
The rate of interest is:
\[ R = 5\% \]

### Final Answer
The rate of interest is **5%**.

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