First nine of that fifteen down!
With 57 games remaining, KC is 6.5 back from Detroit and only 4.5 games back in the wildcard race. Of the remaining 57 games, 49 of those games come either against sub .500 teams or the Tigers or Indians. Basically, the Royals are very close to controlling their own destiny.
30 Games left in August: 15-15 is a realistic basement number
27 Games in September: 13-14 is a realistic basement number
Royals should, at worst, finish with 82 wins, but that is looking at their scheduling and assuming they don't win any of the series against the Tigers or Indians, that they don't even split with Boston, and they don't sweep any series from here on out (in which they still have a few series remaining against both the Twins and White Sox, as well as a series against the Marlins).
Most years, the second team in the wildcard race has between 88 and 91 wins. As of today, the Indians are on pace to win 90 games, Baltimore on pace for 88. The Royals definitely have a shot at 88 if they continue to play like they've played most of this year, aside from a very ugly month of May. And, if they play like they, they'll be taking some wins away from Cleveland.