To determine which of the options provided is a rational number between 5 and 514, let's evaluate each option:
-
517: This number is greater than 514, so it does not fit in the range.
-
\( 5 \frac{1}{7} \): This is a mixed number that can be converted to an improper fraction:
\[
5 \frac{1}{7} = 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14
\]
This is greater than 5 and less than 514.
-
513: This number is less than 514 but greater than 5, so it fits into the range.
-
\( 5 \frac{1}{3} \): This is another mixed number:
\[
5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \approx 5.33
\]
This is also greater than 5 and less than 514.
-
\( \sqrt{26} \): This number is approximately \( 5.099 \) (since \( 5^2 = 25 < 26 < 36 = 6^2 \)). This fits in the range between 5 and 514.
-
525: This number is greater than 514, so it does not fit in the range.
-
\( 5 \frac{2}{5} \): This mixed number can be converted as follows:
\[
5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} \approx 5.4
\]
This is also greater than 5 and less than 514.
Conclusion
The rational numbers between 5 and 514 from the options provided are:
- \( 5 \frac{1}{7} \) (approximately \( 5.14 \))
- \( 513 \)
- \( 5 \frac{1}{3} \) (approximately \( 5.33 \))
- \( \sqrt{26} \) (approximately \( 5.099 \))
- \( 5 \frac{2}{5} \) (approximately \( 5.4 \))
Thus, the rational numbers that fit the criteria are: \( 5 \frac{1}{7} \), 513, \( 5 \frac{1}{3} \), \( \sqrt{26} \), and \( 5 \frac{2}{5} \).
Choose any of these to satisfy the condition of being a rational number between 5 and 514.