After 1 day, the family drank 24 bottles * 1/3 = <<24*1/3=8>>8 bottles
Thus, 24 bottles - 8 = <<24-8=16>>16 remain
On the 2nd day they drank half of the remaining 16, so half would be leftover at the end of day 2, or 16 * 1/2 = <<16*1/2=8>>8 bottles
#### 8