A sum amounts to ₹11,616 in 2 years and to ₹12,777.60 in 3 years, when the interest is compounded anually. The sum (in ₹) is:

Options

A. 8500
B. 8400
C. 9600
D. 9000

Answer

To find the principal sum (P) that amounts to ₹11,616 in 2 years and ₹12,777.60 in 3 years with compound interest, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal amount (initial sum)
- \( r \) = Rate of interest per annum
- \( t \) = Time in years

### Step 2: Set Up the Equations
From the problem, we have two amounts:
1. After 2 years:
\[ A_2 = 11616 = P \left(1 + \frac{r}{100}\right)^2 \]
2. After 3 years:
\[ A_3 = 12777.60 = P \left(1 + \frac{r}{100}\right)^3 \]

### Step 3: Divide the Two Equations
To eliminate \( P \), we can divide the second equation by the first:
\[ \frac{A_3}{A_2} = \frac{P \left(1 + \frac{r}{100}\right)^3}{P \left(1 + \frac{r}{100}\right)^2} \]
This simplifies to:
\[ \frac{12777.60}{11616} = 1 + \frac{r}{100} \]

### Step 4: Calculate the Ratio
Now, calculate the left side:
\[ \frac{12777.60}{11616} \approx 1.1 \]
Thus, we have:
\[ 1.1 = 1 + \frac{r}{100} \]

### Step 5: Solve for \( r \)
Subtract 1 from both sides:
\[ 0.1 = \frac{r}{100} \]
Multiply both sides by 100:
\[ r = 10\% \]

### Step 6: Substitute \( r \) Back into One of the Equations
Now that we have \( r \), we can substitute it back into the first equation:
\[ 11616 = P \left(1 + \frac{10}{100}\right)^2 \]
This simplifies to:
\[ 11616 = P \left(1.1\right)^2 \]
Calculating \( (1.1)^2 \):
\[ (1.1)^2 = 1.21 \]
Thus, we have:
\[ 11616 = P \cdot 1.21 \]

### Step 7: Solve for \( P \)
Now, divide both sides by 1.21:
\[ P = \frac{11616}{1.21} \]
Calculating this gives:
\[ P \approx 9600 \]

### Final Answer
The principal sum (P) is approximately ₹9600.

---