15 Answers

1/3, 2/5, 2/7, 3/7, 3/8, 4/9, 3/10, 3/11, 4/11, 5/11, 5/12, 4/13, 5/13, 6/13, 5/14, 4/15, 7/15, 5/16, 7/16, 5/17, 6/17, 7/17, 8/17, 5/18, 7/18, 5/19, 6/19, 7/19, 8/19, 9/19, 7/20, 9/20.
I tried to make it so that no two fractions were equal to each other, and none were exactly 1/4 or 1/2. Just keep adding 1 to the denominator, and see how many fit between 1/4 and 1/2, that don’t have equivalents that have been used before. So, I didn’t list some fractions, which would have been equivalent to fractions with a lower denominator already listed.

By between, do you mean exactly halfway between or within the interval [1/4, 1/2]
because if it’s the latter, there are an infinite number of ’em.
If you are looking for a fraction a/b exactly halfway between p/q and r/s say. then
a/b = (1/2) * (ps + qr)/qs
So, with p/q = 1/4 and r/s = 1/2
a/b = (1/2) * [(1*2 + 4*1) / (4*2)]
= (1/2)*[(2 + 4)/8]
= (1/2)(6/8)
= 6/16
= 3/8

1/3

1/3

1 / 3

3/8

3/8

3/8

There are infinity possible answers. 1/3 and 3/8 have already been suggested, but there are infinity others.

3/8.