Let's evaluate the question step by step.
To find out how many rational numbers exist between [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex], we need to recall the property of rational numbers. By definition, a rational number is any number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers [tex]\(p\)[/tex] and [tex]\(q\)[/tex], with [tex]\(q \ne 0\)[/tex].
Between any two distinct rational numbers, no matter how close they are, there are infinitely many rational numbers. This is because you can always find another rational number between any two given rational numbers by averaging them or using other methods to generate new fractions.
Given the specific rational numbers [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex], it is possible to find:
- The average [tex]\(\frac{3/5 + 1/5}{2} = \frac{4/5}{2} = \frac{2}{5}\)[/tex], which lies between [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex].
- The midpoint of any two other resulting numbers from these calculations will give additional rational numbers in between.
Because of this, the correct answer should reflect the infinite nature of rational numbers between any two given rational numbers. Therefore, the number of rational numbers between [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex] is many.
The correct option number is:
```
3
```
