To determine which of the provided options is a rational number between 5 and 514, let's convert all options to improper fractions or decimal form for easier comparison.
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\( \sqrt{26} \): This is an irrational number, approximately equal to 5.099. However, irrational numbers are not considered rational.
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\( 5 \frac{1}{8} \): As an improper fraction, this is equal to \( \frac{41}{8} \), which is equal to 5.125. This number is between 5 and 514.
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\( 5 \frac{2}{5} \): As an improper fraction, this is equal to \( \frac{27}{5} \), which is equal to 5.4. This number is also between 5 and 514.
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\( 5 \frac{1}{3} \): As an improper fraction, this is equal to \( \frac{16}{3} \), which is approximately equal to 5.333. This number is also between 5 and 514.
Based on the above assessments, all three numbers (\( 5 \frac{1}{8} \), \( 5 \frac{2}{5} \), and \( 5 \frac{1}{3} \)) are rational numbers between 5 and 514.
If you're looking for just one answer, any of the fractional mixed numbers listed in the last three options are acceptable:
- \( 5 \frac{1}{8} \)
- \( 5 \frac{2}{5} \)
- \( 5 \frac{1}{3} \)
All meet the criteria of being a rational number between 5 and 514.