To find a rational number between \(5\) and \(5 \frac{1}{4}\), we first need to express \(5 \frac{1}{4}\) as an improper fraction:
\[ 5 \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4} \]
Now we are looking for a rational number that is greater than \(5\) (which can also be written as \(\frac{20}{4}\)) and less than \(\frac{21}{4}\).
Next, let's evaluate the given options:
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\( \sqrt{26} \approx 5.099 \) (not a rational number).
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\(5 \frac{2}{5} = 5.4 = \frac{27}{5} = \frac{108}{20} = 5.25\) (this is greater than 5 and less than \(5 \frac{1}{4}\)).
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\(5 \frac{1}{7} = 5.142857 \) (this is greater than \(5\) and less than \(5 \frac{1}{4}\)).
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\(5 \frac{1}{3} = 5.3333\) (this is also greater than \(5 \frac{1}{4}\)).
Thus, the rational numbers between \(5\) and \(5 \frac{1}{4}\) are \(5 \frac{2}{5}\) and \(5 \frac{1}{7}\).
So the rational numbers between \(5\) and \(5 \frac{1}{4}\) are:
- \(5 \frac{2}{5}\)
- \(5 \frac{1}{7}\)
Both options 2 and 3 are valid, but since you're only asking for one, either would suffice. However, \(5 \frac{2}{5}\) is the simplest of the choices to express as a rational number in the specified range.
Answer: \(5 \frac{2}{5}\)