Dana, Gem and Victoria each had some beads and decided to play a game with their beads. In round 1, Dana lost ¹₃ of her beads to Gem. In round 2, Gem lost ¹₃ of her total number of beads to Victoria. In round 3, Victoria lost ¹₃ of her total number of beads to Dana. In the end, Dana, Gem and Victoria had 132, 90 and 68 beads respectively. How many beads did Gem have at first?

	Dana	Gem	Victoria
		86
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	135
Before 2	98	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	102
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Dana, Gem and Victoria in the end	132	90	68

Working backwards.

2 p = 68

1 p = 68 ÷ 2 = 34

At the start of Round 3:
Number of beads that Victoria had
= 3 p
= 3 x 34
= 102

2 u = 90

1 u = 90 ÷ 2 = 45

At the start of Round 2:
Number of beads that Gem had
= 3 u
= 3 x 45
= 135

At the start of Round 2:
Number of beads that Dana had
= 132 - 34
= 98

2 b = 98

1 b = 98 ÷ 2 = 49

At the start of Round 1:
Number of beads that Gem had
= 135 - 1 b
= 135 - 49
= 86

Answer(s): 86