Ivory, Roshel and Tina each had some beads and decided to play a game with their beads. In round 1, Ivory lost ¹₃ of her beads to Roshel. In round 2, Roshel lost ¹₃ of her total number of beads to Tina. In round 3, Tina lost ¹₃ of her total number of beads to Ivory. In the end, Ivory, Roshel and Tina had 139, 90 and 62 beads respectively. How many beads did Roshel have at first?

	Ivory	Roshel	Tina
		81
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	135
Before 2	108	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	93
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Ivory, Roshel and Tina in the end	139	90	62

Working backwards.

2 p = 62

1 p = 62 ÷ 2 = 31

At the start of Round 3:
Number of beads that Tina had
= 3 p
= 3 x 31
= 93

2 u = 90

1 u = 90 ÷ 2 = 45

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

At the start of Round 2:
Number of beads that Ivory had
= 139 - 31
= 108

2 b = 108

1 b = 108 ÷ 2 = 54

At the start of Round 1:
Number of beads that Roshel had
= 135 - 1 b
= 135 - 54
= 81

Answer(s): 81