Gwen, Wendy and Usha each had some beads and decided to play a game with their beads. In round 1, Gwen lost ¹₃ of her beads to Wendy. In round 2, Wendy lost ¹₃ of her total number of beads to Usha. In round 3, Usha lost ¹₃ of her total number of beads to Gwen. In the end, Gwen, Wendy and Usha had 130, 100 and 56 beads respectively. How many beads did Wendy have at first?

	Gwen	Wendy	Usha
		99
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	150
Before 2	102	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	84
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Gwen, Wendy and Usha in the end	130	100	56

Working backwards.

2 p = 56

1 p = 56 ÷ 2 = 28

At the start of Round 3:
Number of beads that Usha had
= 3 p
= 3 x 28
= 84

2 u = 100

1 u = 100 ÷ 2 = 50

At the start of Round 2:
Number of beads that Wendy had
= 3 u
= 3 x 50
= 150

At the start of Round 2:
Number of beads that Gwen had
= 130 - 28
= 102

2 b = 102

1 b = 102 ÷ 2 = 51

At the start of Round 1:
Number of beads that Wendy had
= 150 - 1 b
= 150 - 51
= 99

Answer(s): 99