To find a rational number between \( 2 \frac{1}{4} \) and \( 2 \) using the method of adding the numbers together and dividing by 2, we first need to convert \( 2 \frac{1}{4} \) into an improper fraction.
\[ 2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]
Now we have the two numbers:
- \( \frac{9}{4} \) (which is \( 2 \frac{1}{4} \))
- \( 2 = \frac{2}{1} = \frac{8}{4} \)
Next, we add these two fractions together and divide by 2:
\[ \text{Sum} = \frac{9}{4} + \frac{8}{4} = \frac{17}{4} \]
Now, we divide by 2:
\[ \frac{17}{4} \div 2 = \frac{17}{4} \times \frac{1}{2} = \frac{17}{8} \]
Now, we convert \( \frac{17}{8} \) into a mixed number.
To do this, we divide 17 by 8:
- 8 goes into 17 two times (since \( 2 \times 8 = 16 \)), leaving a remainder of 1.
Thus, we can express \( \frac{17}{8} \) as:
\[ 2 \frac{1}{8} \]
So, a rational number between \( 2 \frac{1}{4} \) and \( 2 \) is
\[ \boxed{2 \frac{1}{8}} \]