if the pairs are denoted A(1)A(2) and B(1)B(2) then A(1)*B(1) must equal A(2)*B(2) where A(1), A(2), B(1), and B(2) can only be positive integer less than 10 (1 to 9). Those integers can produce the ratio 1/2 1/3 2/3 1/4 and 3/4 where A(1)A(2) and B(1)B(2) can be created
ratio 1/2 : (1,2) (2,4) (3,6) (4,8)
ratio 1/3 : (1,3) (2,6) (3,9)
ratio 1/4 : (1,4) (2,8)
ratio 2/3 : (2,3) (4,6) (6,9)
ratio 3/4 : (3,4) (6,8)

Then, we can built a pair of number by selecting 2 members within each ratio group
ratio 1/2 : select 2 out of 4 = 4C2 = 6 ways
ratio 1/3 : select 2 out of 3 = 3C2 = 3 ways
ratio 1/4 : select 2 out of 2 = 2C2 = 1 way
ratio 2/3 : select 2 out of 3 = 3C2 = 3 ways
ratio 3/4 : select 2 out of 2 = 2C2 = 1 way

14 ways in total and they are ...

12 42
12 63
12 84
13 62
13 93
14 82
23 64
23 96
24 63
24 84
26 93
34 86
36 84
46 96

:)