If the amount is `2.25` times of the sum after 2 years at compound Interest (compound annually), the rate of interest per annum is :

Options

A. `25% `
B. `30% `
C. `45% `
D. `50%`

Answer

To solve the problem step by step, we need to find the rate of interest per annum given that the amount is 2.25 times the principal after 2 years of compound interest.

### Step 1: Understand the relationship between principal and amount
Given that the amount (A) is 2.25 times the principal (P) after 2 years, we can express this as:
\[ A = 2.25P \]

### Step 2: Express 2.25 in a simpler form
We can convert 2.25 into a fraction:
\[ 2.25 = \frac{225}{100} = \frac{9}{4} \]
So, we can write:
\[ A = \frac{9}{4}P \]

### Step 3: Use the formula for compound interest
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) is the amount after \( n \) years,
- \( P \) is the principal,
- \( r \) is the rate of interest per annum,
- \( n \) is the number of years.

For our case, \( n = 2 \):
\[ A = P \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Set the two expressions for A equal to each other
Now we set our two expressions for A equal:
\[ \frac{9}{4}P = P \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Cancel P from both sides
Assuming \( P \neq 0 \), we can cancel \( P \):
\[ \frac{9}{4} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 6: Take the square root of both sides
Taking the square root of both sides gives:
\[ \sqrt{\frac{9}{4}} = 1 + \frac{r}{100} \]
\[ \frac{3}{2} = 1 + \frac{r}{100} \]

### Step 7: Solve for r
Now, we solve for \( r \):
\[ \frac{3}{2} - 1 = \frac{r}{100} \]
\[ \frac{1}{2} = \frac{r}{100} \]
Multiplying both sides by 100:
\[ r = 50 \]

### Conclusion
The rate of interest per annum is:
\[ \boxed{50\%} \]

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