Question

How many fractions can we find between `1/3` and `1/4`?

Answer 1

There are infinite number of fractions between `1/3` and `1/4` like `7/24` and `2/7`.

Explanation:

There are infinite fractions between any two fractions. Let the fractions be `a/b` and `c/d`.

Then picking up any two positive numbers `m` and `n`, the number `(m×a/b+n×c/d)/(m+n)` always lies between `a/b` and `c/d`.

Further `(m×a+n×c)/(m×b+n×d)` also lie between `a/b` and `c/d`.

For example choosing `m=1` and `n=1`, the number between `1/3` and `1/4` using first method is `(1/3+1/4)/(1+1)=7/24`, which lies between `1/3` and `1/4`. Similarly using second method `(1+1)/(3+4)=2/7` lies between `1/3` and `1/4`.