got the answer to my earlier quiz.

 19/6 is approximately √10: 10 * 6² = 360; 19² = 361
  9/2 is approximately √20: 20 * 2² = 80; 9² = 81
 11/2 is approximately √30: 30 * 2² = 120; 11² = 121
 19/3 is approximately √40: 40 * 3² = 360; 19² = 361
 99/14 is approximately √50: 50 * 14² = 9800; 99² = 9801
 31/4 is approximately √60: 60 * 4² = 960; 31² = 961
251/30 is approximately √70: 70 * 30² = 63000; 251² = 63001
161/18 is approximately √80: 80 * 18² = 25920; 161² = 25921

So the next in the sequence is 19/2:

 19/2 is approximately √90: 90 * 2² = 360; 19² = 361

points out, reasonably enough, that 9/1 is as good as 161/18 for the eighth fraction in the sequence.