Kimberly, Gwen and Tiffany each had some beads and decided to play a game with their beads. In round 1, Kimberly lost ¹₃ of her beads to Gwen. In round 2, Gwen lost ¹₃ of her total number of beads to Tiffany. In round 3, Tiffany lost ¹₃ of her total number of beads to Kimberly. In the end, Kimberly, Gwen and Tiffany had 126, 86 and 80 beads respectively. How many beads did Gwen have at first?

	Kimberly	Gwen	Tiffany
		86
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	129
Before 2	86	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 Kimberly, Gwen and Tiffany in the end	126	86	80

Working backwards.

2 p = 80

1 p = 80 ÷ 2 = 40

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

2 u = 86

1 u = 86 ÷ 2 = 43

At the start of Round 2:
Number of beads that Gwen had
= 3 u
= 3 x 43
= 129

At the start of Round 2:
Number of beads that Kimberly had
= 126 - 40
= 86

2 b = 86

1 b = 86 ÷ 2 = 43

At the start of Round 1:
Number of beads that Gwen had
= 129 - 1 b
= 129 - 43
= 86

Answer(s): 86