A sum of `Rs. 2500` becomes `Rs. 8100` in 2 years at a certain rate of compound interest. What will be the sum (in Rs.) after 4 years?

Options

A. 29824
B. 36284
C. 41624
D. 26244

Answer

To solve the problem of finding the sum after 4 years given that a principal amount of Rs. 2500 becomes Rs. 8100 in 2 years at a certain rate of compound interest, we can follow these steps:

### Step 1: Understand the Problem
We are given:
- Principal (P) = Rs. 2500
- Amount after 2 years (A) = Rs. 8100
- Time (T) = 2 years

We need to find the amount after 4 years.

### Step 2: Calculate the Compound Interest Factor
The formula for the amount in compound interest is:
\[ A = P(1 + r)^t \]
Where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal
- \( r \) = Rate of interest (as a decimal)
- \( t \) = Time in years

From the information provided:
\[ 8100 = 2500(1 + r)^2 \]

### Step 3: Solve for \( (1 + r)^2 \)
Rearranging the equation to find \( (1 + r)^2 \):
\[ (1 + r)^2 = \frac{8100}{2500} \]
\[ (1 + r)^2 = 3.24 \]

### Step 4: Calculate \( (1 + r) \)
Taking the square root of both sides:
\[ 1 + r = \sqrt{3.24} \]
\[ 1 + r = 1.8 \]
Thus,
\[ r = 1.8 - 1 = 0.8 \text{ or } 80\% \]

### Step 5: Calculate the Amount after 4 Years
Now, we can calculate the amount after 4 years using the formula:
\[ A = P(1 + r)^t \]
For \( t = 4 \):
\[ A = 2500(1 + 0.8)^4 \]
\[ A = 2500(1.8)^4 \]

### Step 6: Calculate \( (1.8)^4 \)
Calculating \( (1.8)^4 \):
\[ (1.8)^4 = 10.4976 \]
Now substituting back:
\[ A = 2500 \times 10.4976 \]
\[ A = 26244 \]

### Step 7: Final Amount
Thus, the amount after 4 years is approximately Rs. 26244.

### Summary of Steps:
1. Understand the problem and identify given values.
2. Use the compound interest formula to find the factor.
3. Solve for \( (1 + r) \).
4. Use the factor to calculate the amount after 4 years.