To determine which of the given numbers is a rational number between 5 and \(5 \frac{1}{4}\) (which is equivalent to \(5.25\)), let's evaluate each option:
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\( \sqrt{26} \): This is an irrational number, as it cannot be expressed as a fraction of two integers.
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\( 5 \frac{1}{8} \): This is equivalent to \(5.125\). Since \(5 < 5.125 < 5.25\), this number is indeed between 5 and \(5 \frac{1}{4}\).
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\( 5 \frac{2}{5} \): This is equivalent to \(5.4\). Since \(5.4\) is greater than \(5.25\), this number is not between 5 and \(5 \frac{1}{4}\).
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\( 5 \frac{1}{3} \): This is equivalent to \(5.333...\). Since \(5.333...\) is also greater than \(5.25\), this number is not between 5 and \(5 \frac{1}{4}\).
The only rational number between 5 and \(5 \frac{1}{4}\) is \(5 \frac{1}{8}\).