A rational number is a number that can be expressed as the quotient or fraction of two integers (i.e., in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \)).
Let's evaluate the options given between 5 and 514:
- 525: This is greater than 514.
- \( 5 \frac{2}{5} \): This can be expressed as \( 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} = 5.4\), which is between 5 and 514.
- 517: This is greater than 514.
- \( 5 \frac{1}{7} \): This can be expressed as \( 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14 \). This is also between 5 and 514.
- \( \sqrt{26} \): This is not a rational number (and it is approximately 5.1), but we need rational numbers specifically.
- 513: This is less than 514 but greater than 5.
The rational numbers between 5 and 514 from these options are \( 5 \frac{2}{5} \) and \( 5 \frac{1}{7} \).
Thus, the correct responses that are rational numbers between 5 and 514 are:
- \( 5 \frac{2}{5} \)
- \( 5 \frac{1}{7} \)
If you are looking for just one answer, \( 5 \frac{2}{5} \) is one valid option.