To find a rational number between 2 and \(2 \frac{1}{4}\) using the method of adding the numbers together and dividing by two, we start by converting \(2 \frac{1}{4}\) into an improper fraction.
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Convert \(2 \frac{1}{4}\) to an improper fraction: \[ 2 \frac{1}{4} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]
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Now we want to find a number between 2 and \(\frac{9}{4}\). First, we convert 2 into a fraction with a denominator of 4: \[ 2 = \frac{8}{4} \]
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Add the two fractions: \[ \frac{8}{4} + \frac{9}{4} = \frac{17}{4} \]
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Divide by 2 to find the average: \[ \frac{17}{4} \div 2 = \frac{17}{4} \times \frac{1}{2} = \frac{17}{8} \]
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Convert \(\frac{17}{8}\) to a mixed number:
- Divide 17 by 8, which goes 2 times (as \(2 \times 8 = 16\)).
- The remainder is \(17 - 16 = 1\).
- So, \(\frac{17}{8} = 2 \frac{1}{8}\).
Thus, a rational number between 2 and \(2 \frac{1}{4}\) is:
\[ 2 \frac{1}{8} \]