To find a rational number between \(2\) and \(2 \frac{1}{4}\), we can use the method of adding the two numbers together and then dividing by \(2\).
First, we convert \(2 \frac{1}{4}\) into an improper fraction.
\[ 2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]
Next, we add \(2\) (which is \(\frac{8}{4}\)) and \(\frac{9}{4}\):
\[ \frac{8}{4} + \frac{9}{4} = \frac{8 + 9}{4} = \frac{17}{4} \]
Now, we divide this sum by \(2\):
\[ \frac{17}{4} \div 2 = \frac{17}{4} \times \frac{1}{2} = \frac{17}{8} \]
Our next step is to express \(\frac{17}{8}\) as a mixed number.
To do this, we divide \(17\) by \(8\):
\[ 17 \div 8 = 2 \quad \text{remainder } 1 \]
Thus, we can write:
\[ \frac{17}{8} = 2 \frac{1}{8} \]
Therefore, a rational number between \(2\) and \(2 \frac{1}{4}\) is:
\[ \boxed{2 \frac{1}{8}} \]
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