The total beads never change, so split it into 25 equal units. Lucy is $$\frac{1}{4}$$ = 5 units before (1 : 4) and $$\frac{6}{19}$$ = 6 units after (6 : 19). The 25 beads she received are the extra 1 unit.<br><br>[[model:questions/solutions/q5291.png]]<br><br>• Lucy’s gain → 6 units − 5 units = 1 unit = 25 beads<br>• 1 unit → 25<br>• Lucy at first → 5 units = 5 × 25 = 125<br><br>Lucy had 125 beads at first.<br>Check: total 25 units = 625; before Lucy 125, Beth 500 (1 : 4); after Lucy 150, Beth 475 = 6 : 19. ✓