The compound interest on a certain sum at a certain rate percent per annum for the second year and the third year are ₹3300 and ₹3630, respectively. The sum is:

Options

A. ₹32000
B. ₹28400
C. ₹30000
D. ₹25000

Answer

To solve the problem, we need to determine the principal amount based on the compound interest for the second and third years. Let's break it down step by step.

### Step 1: Understand the given information
We know that:
- Compound Interest (CI) for the second year = ₹3300
- Compound Interest (CI) for the third year = ₹3630

### Step 2: Calculate the difference in CI
To find the rate of interest, we first calculate the difference between the CI for the third year and the second year.

\[ \text{Difference} = \text{CI for 3rd year} - \text{CI for 2nd year} = 3630 - 3300 = 330 \]

### Step 3: Relate the difference to the principal and rate
The difference in CI between the second and third years is due to the interest applied on the principal amount. This difference represents the interest earned on the principal for the second year.

Let the rate of interest be \( r \) (in percentage). The difference can be expressed as:

\[ \text{Difference} = \frac{r}{100} \times \text{Principal} \]

From the previous calculation, we know that the difference is ₹330.

### Step 4: Express the principal in terms of the rate
Rearranging the equation gives us:

\[ \text{Principal} = \frac{330 \times 100}{r} \]

### Step 5: Determine the rate of interest
From the CI for the second year (₹3300), we can also express it in terms of the principal and the rate:

\[ \text{CI for 2nd year} = \text{Principal} \times \frac{r}{100} \]

Substituting the value of CI for the second year:

\[ 3300 = \text{Principal} \times \frac{r}{100} \]

### Step 6: Substitute the principal value
Now we have two equations:
1. \( \text{Principal} = \frac{330 \times 100}{r} \)
2. \( 3300 = \text{Principal} \times \frac{r}{100} \)

Substituting the first equation into the second:

\[ 3300 = \left(\frac{330 \times 100}{r}\right) \times \frac{r}{100} \]

### Step 7: Simplify the equation
This simplifies to:

\[ 3300 = 330 \]

This means that the rate \( r \) must be 10% (since \( 3300 = 330 \times 10 \)).

### Step 8: Calculate the principal
Now substituting \( r = 10 \) back into the principal equation:

\[ \text{Principal} = \frac{330 \times 100}{10} = 3300 \]

### Step 9: Find the correct principal amount
To find the principal amount, we can also use the CI for the first year:

\[ \text{CI for 1st year} = \text{Principal} \times \frac{10}{100} = 3000 \]

Thus, the principal amount is:

\[ \text{Principal} = 3000 \times 10 = 30,000 \]

### Conclusion
The principal amount is ₹30,000.