Step 1: Write the Compound Interest Formula

The formula for the amount $A$ after $n$ years at an annual compound interest rate $R$ is given by $A = P(1 + R/100)^n$.

$A = P \left(1 + \frac{R}{100}\right)^n$

Step 2: Substitute the Given Values

We are given that the sum amounts to $2\frac{1}{4}$ times itself in $2$ years. Thus, $A = \frac{9}{4}P$ and $n = 2$.

$\frac{9}{4}P = P \left(1 + \frac{R}{100}\right)^2$

Step 3: Simplify and Take the Square Root

Dividing both sides by $P$ and taking the square root since time $n = 2$:

$\frac{9}{4} = \left(1 + \frac{R}{100}\right)^2 \implies \frac{3}{2} = 1 + \frac{R}{100}$

Step 4: Calculate the Rate of Interest

Solving the linear equation for $R$ gives the annual rate of interest.

$\frac{R}{100} = \frac{3}{2} - 1 = \frac{1}{2} \implies R = 50\%$