A sum of Rs. 5120 amounts to Rs. 7290 in 3 years at compound interest. Find the rate of interest per annum.

Options

A. `33(1)/(3)%`
B. `12(1)/(2)%`
C. `8(1)/(3)%`
D. `17(1)/(2)%`

Explanation

To find the rate of interest per annum when a sum of Rs. 5120 amounts to Rs. 7290 in 3 years at compound interest, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 5120
- Amount (A) = Rs. 7290
- Time (T) = 3 years

### Step 2: 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 = Amount
- P = Principal
- R = Rate of interest per annum
- T = Time in years

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 7290 = 5120 \left(1 + \frac{R}{100}\right)^3 \]

### Step 4: Divide both sides by the principal
To isolate the term with R, divide both sides by 5120:
\[ \frac{7290}{5120} = \left(1 + \frac{R}{100}\right)^3 \]

### Step 5: Calculate the left side
Now calculate the left side:
\[ \frac{7290}{5120} = 1.42578125 \]

### Step 6: Take the cube root of both sides
Next, take the cube root of both sides to solve for \(1 + \frac{R}{100}\):
\[ 1 + \frac{R}{100} = \sqrt[3]{1.42578125} \]

### Step 7: Calculate the cube root
Calculating the cube root:
\[ 1 + \frac{R}{100} \approx 1.110 \]

### Step 8: Solve for R
Now, subtract 1 from both sides:
\[ \frac{R}{100} \approx 0.110 \]
Multiply both sides by 100 to find R:
\[ R \approx 11.0 \]

### Step 9: Conclusion
The rate of interest per annum is approximately **11%**.

---