Kylie, Lynn and Xuan each had some beads and decided to play a game with their beads. In round 1, Kylie lost ¹₃ of her beads to Lynn. In round 2, Lynn lost ¹₃ of her total number of beads to Xuan. In round 3, Xuan lost ¹₃ of her total number of beads to Kylie. In the end, Kylie, Lynn and Xuan had 146, 74 and 80 beads respectively. How many beads did Lynn have at first?

	Kylie	Lynn	Xuan
		58
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	111
Before 2	106	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	120
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Kylie, Lynn and Xuan in the end	146	74	80

Working backwards.

2 p = 80

1 p = 80 ÷ 2 = 40

At the start of Round 3:
Number of beads that Xuan had
= 3 p
= 3 x 40
= 120

2 u = 74

1 u = 74 ÷ 2 = 37

At the start of Round 2:
Number of beads that Lynn had
= 3 u
= 3 x 37
= 111

At the start of Round 2:
Number of beads that Kylie had
= 146 - 40
= 106

2 b = 106

1 b = 106 ÷ 2 = 53

At the start of Round 1:
Number of beads that Lynn had
= 111 - 1 b
= 111 - 53
= 58

Answer(s): 58