A certain sum amounts to ₹ 8988.8 in two years and to ₹ 9528.128 in three years, at compound interest per annum. What is the principal and rate of interest?

Options

A. `₹1200,5%`
B. `₹ 6,00,8%`
C. `₹ 8,000,6%`
D. `₹ 10,000,8.5%`

Answer

To solve the problem, we need to find the principal amount (P) and the rate of interest (R) given the amounts after 2 years and 3 years at compound interest.

### Step 1: Understand the relationship between amounts
Let:
- A1 = Amount after 2 years = ₹ 8988.8
- A2 = Amount after 3 years = ₹ 9528.128

The relationship between the amounts can be expressed as:
\[ A2 = A1 \times (1 + R) \]

### Step 2: Set up the equation
From the relationship above, we can express the rate of interest (R) as:
\[ 9528.128 = 8988.8 \times (1 + R) \]

### Step 3: Solve for (1 + R)
Rearranging the equation gives:
\[ 1 + R = \frac{9528.128}{8988.8} \]

### Step 4: Calculate (1 + R)
Now, calculate the value:
\[ 1 + R = \frac{9528.128}{8988.8} \approx 1.0596 \]

### Step 5: Find R
Now, subtract 1 from both sides to find R:
\[ R \approx 1.0596 - 1 = 0.0596 \]
\[ R \approx 0.0596 \times 100 = 5.96\% \]

### Step 6: Use A1 to find the principal
Now that we have R, we can find the principal amount using the formula for the amount after 2 years:
\[ A1 = P \times (1 + R)^2 \]
Substituting the known values:
\[ 8988.8 = P \times (1.0596)^2 \]

### Step 7: Calculate (1 + R)^2
Calculating \( (1.0596)^2 \):
\[ (1.0596)^2 \approx 1.1275 \]

### Step 8: Solve for P
Now we can rearrange the equation to find P:
\[ P = \frac{8988.8}{1.1275} \approx 7976.23 \]

### Final Results
- Principal (P) = ₹ 7976.23
- Rate of Interest (R) = 5.96%