Sonika invested an amount of Rs.5800 for 2 years. At what rate of compound interest will she get an amount of Rs.594.50 at the end of 2 years ?

Options

A. A. 5%
B. B. 11%
C. B. 20%
D. D. 23%

Explanation

To solve the problem step by step, we need to find the rate of compound interest at which Sonika will earn Rs. 594.50 on her investment of Rs. 5800 over 2 years.

### Step 1: Identify the given values
- Principal (P) = Rs. 5800
- Compound Interest (CI) = Rs. 594.50
- Time (T) = 2 years

### Step 2: Calculate the total amount (A) at the end of 2 years
The total amount is calculated by adding the principal and the compound interest:
\[ A = P + CI \]
Substituting the values:
\[ A = 5800 + 594.50 = 6394.50 \]

### Step 3: Use the formula for compound interest
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- A = Total amount after time T
- P = Principal
- R = Rate of interest
- T = Time in years

### Step 4: Substitute the known values into the formula
Substituting the values we have:
\[ 6394.50 = 5800 \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Rearranging the equation
To isolate the term with R, we divide both sides by 5800:
\[ \frac{6394.50}{5800} = \left(1 + \frac{R}{100}\right)^2 \]
Calculating the left side:
\[ \frac{6394.50}{5800} = 1.1025 \]

### Step 6: Take the square root of both sides
Now, we take the square root to solve for \(1 + \frac{R}{100}\):
\[ \sqrt{1.1025} = 1 + \frac{R}{100} \]
Calculating the square root:
\[ 1.05 = 1 + \frac{R}{100} \]

### Step 7: Isolate R
Now, we isolate R:
\[ \frac{R}{100} = 1.05 - 1 \]
\[ \frac{R}{100} = 0.05 \]
Multiplying both sides by 100 gives:
\[ R = 5 \]

### Step 8: Conclusion
The rate of compound interest is **5%**.