A sum of Rs 2000 becomes Rs 3645 in 2 years at a certain rate of compound interest. What will be the sum (in Rs) after 4 years. ?

Options

A. 5942.125
B. 6643.0125
C. 7243.0125
D. 7498.125

Answer

To solve the problem step by step, we will use the concept of compound interest and the relationship between amounts at different time periods.

### Step 1: Understand the Problem
We are given:
- Principal (P) = Rs 2000
- Amount after 2 years (A) = Rs 3645
- We need to find the amount after 4 years.

### Step 2: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P(1 + r)^n \]
where:
- \( A \) is the amount after time \( n \),
- \( P \) is the principal,
- \( r \) is the rate of interest per period,
- \( n \) is the number of periods.

### Step 3: Find the Rate of Interest
From the information given, we can set up the equation for the amount after 2 years:
\[ 3645 = 2000(1 + r)^2 \]

### Step 4: Solve for \( (1 + r)^2 \)
First, divide both sides by 2000:
\[ (1 + r)^2 = \frac{3645}{2000} \]
\[ (1 + r)^2 = 1.8225 \]

### Step 5: Take the Square Root
Now, take the square root of both sides to find \( 1 + r \):
\[ 1 + r = \sqrt{1.8225} \]
Calculating the square root:
\[ 1 + r \approx 1.35 \]

### Step 6: Calculate \( r \)
Now, subtract 1 from both sides:
\[ r \approx 0.35 \]
Thus, the rate of interest \( r \) is approximately 0.35 or 35%.

### Step 7: Find the Amount After 4 Years
Now we need to find the amount after 4 years. We can use the compound interest formula again:
\[ A = P(1 + r)^n \]
Here, \( n = 4 \):
\[ A = 2000(1.35)^4 \]

### Step 8: Calculate \( (1.35)^4 \)
Calculating \( (1.35)^4 \):
\[ (1.35)^4 \approx 2.5735 \]

### Step 9: Calculate the Final Amount
Now substitute back to find \( A \):
\[ A = 2000 \times 2.5735 \]
\[ A \approx 5147 \]

Thus, the amount after 4 years will be approximately Rs 5147.

### Final Answer
The sum after 4 years will be Rs 5147.

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