A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).
Now, let's analyze the given options to see which ones are rational numbers between 5 and 514:
- 513: This is a whole number (and can be expressed as \(\frac{513}{1}\)), and it is between 5 and 514.
- 5: This is a whole number, but it's not between 5 and 514 (it's equal to 5).
- \(\frac{1}{3}\): This is a rational number, but it is not between 5 and 514.
- \(\sqrt{26}\): This is an irrational number (it cannot be expressed as a fraction).
- 525: This is greater than 514, so it doesn't fit.
- \(\frac{2}{5}\): This is a rational number but is not between 5 and 514.
- 517: This is a whole number (and can be expressed as \(\frac{517}{1}\)), and it is between 5 and 514.
- \(\frac{1}{7}\): This is a rational number, but it is not between 5 and 514.
The rational numbers from the choices provided that are between 5 and 514 are 513 and 517.