The rate of interest for a sum that becomes `(729)/(576)` times itself in 2 years, when compounded annually, is

Options

A. `(32)/(5)%`
B. `(12)/(5)%`
C. `(25)/(2)%`
D. `(12)/(7)%`

Explanation

To find the rate of interest for a sum that becomes \( \frac{729}{576} \) times itself in 2 years when compounded annually, we can follow these steps:

### Step 1: Understand the relationship between Principal, Amount, and Rate of Interest
The formula for the amount \( A \) when compounded annually is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where:
- \( A \) is the amount after time \( t \)
- \( P \) is the principal amount (initial sum)
- \( r \) is the rate of interest per annum
- \( t \) is the time in years

### Step 2: Set up the equation
Given that the amount becomes \( \frac{729}{576} \) times the principal \( P \) in 2 years, we can write:
\[ \frac{729}{576} P = P \left(1 + \frac{r}{100}\right)^2 \]

### Step 3: Cancel \( P \) from both sides
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{729}{576} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Take the square root of both sides
To solve for \( 1 + \frac{r}{100} \), we take the square root of both sides:
\[ 1 + \frac{r}{100} = \sqrt{\frac{729}{576}} \]

### Step 5: Simplify the square root
Calculating the square root:
\[ \sqrt{\frac{729}{576}} = \frac{\sqrt{729}}{\sqrt{576}} = \frac{27}{24} \]

### Step 6: Set up the equation to find \( r \)
Now we have:
\[ 1 + \frac{r}{100} = \frac{27}{24} \]

### Step 7: Isolate \( \frac{r}{100} \)
Subtract 1 from both sides:
\[ \frac{r}{100} = \frac{27}{24} - 1 = \frac{27}{24} - \frac{24}{24} = \frac{3}{24} = \frac{1}{8} \]

### Step 8: Solve for \( r \)
Multiply both sides by 100:
\[ r = 100 \times \frac{1}{8} = 12.5 \]

### Conclusion
The rate of interest is \( 12.5\% \) per annum.

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