The difference between the compound interest and simple interest for the amount Rs. 5,000 in 2 years is Rs.32. The rate of interest is

Options

A. `5%`
B. `8%`
C. `10%`
D. `12%`

Answer

To find the rate of interest given that the difference between the compound interest (CI) and simple interest (SI) for an amount of Rs. 5,000 in 2 years is Rs. 32, we can follow these steps:

### Step 1: Understand the formulas
The formulas for compound interest and simple interest are:
- Simple Interest (SI) = \( P \times \frac{R \times T}{100} \)
- Compound Interest (CI) = \( P \left(1 + \frac{R}{100}\right)^T - P \)

Where:
- \( P \) = Principal amount (Rs. 5,000)
- \( R \) = Rate of interest (unknown)
- \( T \) = Time (2 years)

### Step 2: Set up the equation
The difference between CI and SI is given as Rs. 32. Therefore, we can write:
\[ CI - SI = 32 \]

Substituting the formulas:
\[ P \left(1 + \frac{R}{100}\right)^T - P - \left(P \times \frac{R \times T}{100}\right) = 32 \]

### Step 3: Substitute the known values
Substituting \( P = 5000 \) and \( T = 2 \):
\[ 5000 \left(1 + \frac{R}{100}\right)^2 - 5000 - \left(5000 \times \frac{R \times 2}{100}\right) = 32 \]

### Step 4: Simplify the equation
First, simplify the left-hand side:
\[ 5000 \left(1 + \frac{R}{100}\right)^2 - 5000 - \frac{10000R}{100} = 32 \]
\[ 5000 \left(1 + \frac{R}{100}\right)^2 - 5000 - 100R = 32 \]

### Step 5: Factor out 5000
Now, factor out 5000:
\[ 5000 \left(\left(1 + \frac{R}{100}\right)^2 - 1 - \frac{2R}{100}\right) = 32 \]

### Step 6: Divide by 5000
Divide both sides by 5000:
\[ \left(1 + \frac{R}{100}\right)^2 - 1 - \frac{2R}{100} = \frac{32}{5000} \]
\[ \left(1 + \frac{R}{100}\right)^2 - 1 - \frac{2R}{100} = 0.0064 \]

### Step 7: Expand the left-hand side
Now, expand the left-hand side:
\[ \left(1 + \frac{R}{100}\right)^2 = 1 + 2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2 \]
Thus,
\[ 1 + 2 \cdot \frac{R}{100} + \left(\frac{R}{100}\right)^2 - 1 - \frac{2R}{100} = 0.0064 \]
This simplifies to:
\[ \left(\frac{R}{100}\right)^2 = 0.0064 \]

### Step 8: Solve for R
Taking the square root of both sides:
\[ \frac{R}{100} = 0.08 \]
Thus,
\[ R = 0.08 \times 100 = 8 \]

### Conclusion
The rate of interest is **8%**.