Monica deposited a total of Rs. 10500 with a bank in two different deposit schemes at 10% p.a., interest being compounded annually. As per the schemes, she gets the same amount after 2 years on the first deposit as she gets after 3 years on the second deposit. How much money did she deposit for 3 years?

Options

A. A)Rs. 4500
B. B)Rs. 5000
C. C)Rs. 6500
D. D)Rs. 7200

Answer

To solve the problem step by step, we will denote the amounts deposited in the two schemes as follows:

Let:
- \( P_A \) = Amount deposited in scheme A (for 2 years)
- \( P_B \) = Amount deposited in scheme B (for 3 years)

Given:
- Total amount deposited: \( P_A + P_B = 10500 \)
- Rate of interest: \( 10\% \) per annum
- Amount after 2 years in scheme A is equal to the amount after 3 years in scheme B.

### Step 1: Write the formula for compound interest
The formula for the amount \( A \) after \( t \) years with principal \( P \) and rate \( r \) is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]

### Step 2: Write the amount equations for both schemes
For scheme A (2 years):
\[ A_A = P_A \left(1 + \frac{10}{100}\right)^2 = P_A \left(1.1\right)^2 = P_A \cdot 1.21 \]

For scheme B (3 years):
\[ A_B = P_B \left(1 + \frac{10}{100}\right)^3 = P_B \left(1.1\right)^3 = P_B \cdot 1.331 \]

### Step 3: Set the amounts equal to each other
Since the amounts after the respective periods are equal:
\[ P_A \cdot 1.21 = P_B \cdot 1.331 \]

### Step 4: Express \( P_A \) in terms of \( P_B \)
Rearranging the equation gives:
\[ P_A = \frac{1.331}{1.21} P_B \]

### Step 5: Calculate the ratio of \( P_A \) to \( P_B \)
Calculating the ratio:
\[ \frac{P_A}{P_B} = \frac{1.331}{1.21} \approx 1.1 \]
This means:
\[ P_A : P_B = 11 : 10 \]

### Step 6: Express \( P_A \) and \( P_B \) in terms of a single variable
Let \( P_A = 11x \) and \( P_B = 10x \).

### Step 7: Use the total amount equation
Substituting into the total amount equation:
\[ 11x + 10x = 10500 \]
\[ 21x = 10500 \]
\[ x = \frac{10500}{21} = 500 \]

### Step 8: Calculate \( P_B \)
Now, substituting back to find \( P_B \):
\[ P_B = 10x = 10 \times 500 = 5000 \]

### Conclusion
Thus, the amount Monica deposited for the scheme that lasts for 3 years is:
\[ \boxed{5000} \]