The total money is unchanged, so split it into 36 equal units. Before, Sam = 1⁄8 of Patrick = 1⁄9 of the total = 4 units. After, Sam = 1⁄3 of Patrick = 1⁄4 of the total = 9 units. Sam gained 9 − 4 = 5 units.

Sam (before)Patrick (before)Sam (after)Patrick (after)+ $60

Plain numbers are given; italic underlined numbers are worked out.

  1. Sam’s gain → 9 units − 4 units = 5 units → $60
  2. 1 unit → $60 ÷ 5 = $12
  3. Sam in the end → 9 units = 9 × $12 = $108

Sam has $108 in the end.

Check: before Sam $48, Patrick $384 (1:8 ✓); after Sam $108, Patrick $324 = 1:3. ✓