A certain amount of money earns `Rs. 540` as Simple Interest in 3 years. If it earns a Compound Interest of `Rs. 376.20` at the same rate of Interest in 2 years, find the amount (in Rupees).

Options

A. `1600 `
B. `1800 `
C. `2000 `
D. `2100 `

Answer

To solve the problem step by step, we will first extract the necessary information and then use the formulas for Simple Interest (SI) and Compound Interest (CI) to find the required amount.

### Step 1: Understand the given information
- Simple Interest (SI) earned in 3 years = Rs. 540
- Compound Interest (CI) earned in 2 years = Rs. 376.20

### Step 2: Calculate the rate of interest
From the SI formula:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

We know that:
\[ 540 = \frac{P \times R \times 3}{100} \]
This can be rearranged to:
\[ P \times R = \frac{540 \times 100}{3} = 18000 \quad \text{(1)} \]

### Step 3: Relate CI to SI
The formula for Compound Interest for 2 years is:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^2 - P \]
We can also express CI in terms of SI:
\[ \text{CI} = \text{SI for 2 years} + \text{SI for 1 year} \]
Where:
- SI for 2 years = \( \frac{P \times R \times 2}{100} \)
- SI for 1 year = \( \frac{P \times R \times 1}{100} \)

From the SI formula for 2 years:
\[ \text{SI for 2 years} = \frac{P \times R \times 2}{100} \]
Thus,
\[ \text{CI} = \frac{P \times R \times 2}{100} + \frac{P \times R \times 1}{100} = \frac{P \times R \times 3}{100} \]
Since we already know \( P \times R = 18000 \) from equation (1):
\[ \text{CI} = \frac{18000 \times 3}{100} = 540 \]

### Step 4: Calculate the difference between CI and SI
The difference between CI and SI for 2 years is:
\[ \text{Difference} = \text{CI} - \text{SI for 2 years} \]
Given that CI = Rs. 376.20, we can find SI for 2 years:
\[ \text{SI for 2 years} = \frac{P \times R \times 2}{100} = \frac{18000 \times 2}{100} = 360 \]
Thus,
\[ \text{Difference} = 376.20 - 360 = 16.20 \]

### Step 5: Find the rate of interest
The difference of Rs. 16.20 represents the interest on the interest for one year. Therefore, we can express this as:
\[ \frac{P \times R}{100} = 16.20 \]
Using \( P \times R = 18000 \):
\[ \frac{18000}{100} = 16.20 \]
This gives us:
\[ R = \frac{16.20 \times 100}{P} \]

### Step 6: Substitute and solve for P
From the SI equation:
\[ 540 = \frac{P \times R \times 3}{100} \]
Substituting \( R \):
\[ 540 = \frac{P \times \frac{16.20 \times 100}{P} \times 3}{100} \]
This simplifies to:
\[ 540 = 16.20 \times 3 \]
Calculating gives:
\[ 540 = 48.60 \]
This means \( P = \frac{540 \times 100}{48.60} \approx 1111.11 \)

### Step 7: Find the total amount
The total amount (A) can be calculated using the formula:
\[ A = P + \text{SI} \]
Substituting the values we have:
\[ A = 1111.11 + 540 = 1651.11 \]

### Final Answer
The amount is approximately Rs. 1651.11.