Two nonnegative reduced fractions $a/b$ and $c/d$ make a Farey pair (with $a/b < c/d$ ) whenever $bc-ad=1$ , in other words, they are a Farey pair if their difference is $1/bd$ . The interval $[a/b, c/d]$ is known as a Farey interval.

Given a Farey pair $a/b,c/d$ , their mediant is $(a+c)/(b+d)$ . The mediant has the following property:

If $[a,b,c/d]$ is a Farey interval, then the two subintervals obtained when inserting the mediant are also Farey pairs. Besides, between all fractions that are strictly between $a/b,c/d$ , the mediant is the one having the smallest denominator.

Example.
Notice that $3/8$ and $5/11$ form a Farey pair, since $8\cdot 5 - 3\cdot 13 =40-391$ . The mediant here is $8/21$ .

Then $3/8$ and $8/21$ form a Farey pair: $8\cdot 8 - 3\cdot 21 = 64-63=1$ . No fraction between $3/8$ and $5/11$ other than $8/21$ has a denominator smaller or equal than $21$ .