No beads are created or lost, so the total (Lucy + Beth) stays the same — 14 units. Before, Lucy is 5 units (5⁄9 of Beth’s 9). After, Lucy : Beth = 3 : 4 = 6 : 8 units. Lucy went from 5 units to 6 units, a gain of 1 unit.

Lucy (before)11111Beth (before)111111111Lucy (after)11111+25Beth (after)11111111

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

  1. Lucy’s gain → 6 units − 5 units = 1 unit → 25 beads
  2. 1 unit → 25 beads
  3. Lucy at first → 5 units = 5 × 25 = 125

Lucy had 125 beads at first.

Check: Lucy 125, Beth 225 (5:9 ✓); after, Lucy 150, Beth 200 = 3:4. ✓