To find a rational number between \( 2 \) and \( 2 \frac{1}{4} \) using the method of averaging, we first convert \( 2 \frac{1}{4} \) into an improper fraction.
The mixed number \( 2 \frac{1}{4} \) can be converted as follows:
\[ 2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]
Now we have the two numbers in a form that we can work with: \( 2 \) (which can be expressed as \( \frac{8}{4} \)) and \( \frac{9}{4} \).
Next, we'll find a rational number between these two values by averaging them:
\[ \text{Average} = \frac{2 + 2 \frac{1}{4}}{2} = \frac{\frac{8}{4} + \frac{9}{4}}{2} \]
Now, let's combine the fractions:
\[ \frac{\frac{8}{4} + \frac{9}{4}}{2} = \frac{\frac{17}{4}}{2} = \frac{17}{4} \times \frac{1}{2} = \frac{17}{8} \]
Now, we convert \( \frac{17}{8} \) back into a mixed number:
To convert \( \frac{17}{8} \) into a mixed number, we divide \( 17 \) by \( 8 \):
\[ 17 \div 8 = 2 \quad \text{(the whole number part)} \] \[ 17 - 2 \times 8 = 17 - 16 = 1 \quad \text{(the remainder)} \]
Thus, we have:
\[ \frac{17}{8} = 2 \frac{1}{8} \]
So, a rational number between \( 2 \) and \( 2 \frac{1}{4} \) is \( \boxed{2 \frac{1}{8}} \).