Answer: at most 15 pairs of socks
Step-by-step explanation:
Let the number of pair of socks of her sister be  , and let the number of pairs Lucy has be
 , and let the number of pairs Lucy has be  .
.
Since , Lucy has 4 more than 1/3 the number of pairs of socks as her sister, this means that
 .............................. equation 1
    .............................. equation 1
Also ,  Lucy has at most 9 pairs of socks , this means that
 ....................................... equation 2
 ....................................... equation 2
Equating the two equations , we have 

multiply through by 3

subtract 12 from both sides 

Therefore , the sister has at most 15 pairs of socks