An amount is invested in a bank at compound rate of interest . The total amount, including interest, after first and third year is X 1200 and X 1587 , respecitvely. What is the rate of interest ?

Options

A. 1
B. 39
C. 12
D. 15

Answer

To find the rate of interest based on the given amounts after the first and third years, we can use the formula for compound interest.

### Step-by-step Solution:

1. **Identify the given amounts:**
- Amount after 1st year (A1) = ₹1200
- Amount after 3rd year (A3) = ₹1587

2. **Let the principal amount be P and the rate of interest be R% per annum.**

3. **Using the formula for compound interest:**
The amount after the first year can be expressed as:
\[ A_1 = P \left(1 + \frac{R}{100}\right) \]
Substituting the value of A1:
\[ 1200 = P \left(1 + \frac{R}{100}\right) \quad \text{(1)} \]

4. **For the third year, the amount can be expressed as:**
\[ A_3 = P \left(1 + \frac{R}{100}\right)^3 \]
Substituting the value of A3:
\[ 1587 = P \left(1 + \frac{R}{100}\right)^3 \quad \text{(2)} \]

5. **Dividing equation (2) by equation (1):**
\[ \frac{1587}{1200} = \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)} \]
This simplifies to:
\[ \frac{1587}{1200} = \left(1 + \frac{R}{100}\right)^2 \]

6. **Calculating the left side:**
\[ \frac{1587}{1200} = 1.3225 \]
Thus, we have:
\[ 1.3225 = \left(1 + \frac{R}{100}\right)^2 \]

7. **Taking the square root of both sides:**
\[ 1 + \frac{R}{100} = \sqrt{1.3225} \]
Calculating the square root:
\[ 1 + \frac{R}{100} \approx 1.15 \]

8. **Solving for R:**
\[ \frac{R}{100} = 1.15 - 1 = 0.15 \]
\[ R = 0.15 \times 100 = 15 \]

### Final Answer:
The rate of interest is **15%**.