The difference between simple interest and compound interest on a sum of Rs 40,000 for two years is Rs 900. What is the annual rate of interest?

Options

A. `20%`
B. `10%`
C. `12%`
D. `15%`

Answer

To find the annual rate of interest given that the difference between simple interest (SI) and compound interest (CI) on a sum of Rs 40,000 for two years is Rs 900, we can follow these steps:

### Step 1: Understand the formulas for SI and CI
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times r \times t}{100} \]
- The formula for Compound Interest (CI) is:
\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]
where \( n \) is the number of years.

### Step 2: Set up the equation
Given:
- Principal (P) = Rs 40,000
- Time (t) = 2 years
- Difference between CI and SI = Rs 900

We can express this as:
\[ CI - SI = 900 \]

### Step 3: Substitute the formulas into the equation
Substituting the formulas into the equation gives:
\[ P \left( \left(1 + \frac{r}{100}\right)^2 - 1 \right) - \frac{P \times r \times t}{100} = 900 \]
Substituting \( P = 40,000 \) and \( t = 2 \):
\[ 40,000 \left( \left(1 + \frac{r}{100}\right)^2 - 1 \right) - \frac{40,000 \times r \times 2}{100} = 900 \]

### Step 4: Simplify the equation
This simplifies to:
\[ 40,000 \left( \left(1 + \frac{r}{100}\right)^2 - 1 \right) - 800r = 900 \]
Dividing the entire equation by 40,000:
\[ \left( \left(1 + \frac{r}{100}\right)^2 - 1 \right) - \frac{800r}{40,000} = \frac{900}{40,000} \]
This simplifies to:
\[ \left( \left(1 + \frac{r}{100}\right)^2 - 1 \right) - \frac{r}{50} = \frac{9}{400} \]

### Step 5: Expand the left side
Expanding \( \left(1 + \frac{r}{100}\right)^2 \):
\[ 1 + 2 \cdot \frac{r}{100} + \left(\frac{r}{100}\right)^2 - 1 - \frac{r}{50} = \frac{9}{400} \]
This simplifies to:
\[ \frac{2r}{100} + \frac{r^2}{10,000} - \frac{r}{50} = \frac{9}{400} \]

### Step 6: Find a common denominator and simplify
The common denominator for the left side is 10,000:
\[ \frac{200r}{10,000} + \frac{r^2}{10,000} - \frac{200r}{10,000} = \frac{9}{400} \]
This simplifies to:
\[ \frac{r^2}{10,000} = \frac{9}{400} \]
Cross-multiplying gives:
\[ r^2 = 9 \cdot 25 = 225 \]

### Step 7: Solve for r
Taking the square root:
\[ r = 15 \]

### Conclusion
The annual rate of interest is:
\[ \text{Rate of Interest} = 15\% \]