Any number that can be written as a fraction is rational! Since both 1/8 and 1/4 have the same numerator it is easy to compare them and see that:
1/5, 1/6, and 1/7 are all examples or rational numbers in between them.
You could also add both 1/8 and 1/4 and divide by 2 to get 3/16, which is also a rational number in between them.
Of course, these are examples of methods you could use over and over again, but there are also infinite more possible answers to your question.