Risa, Gwen and Opal each had some balls and decided to play a game with their balls. In round 1, Risa lost ¹₃ of her balls to Gwen. In round 2, Gwen lost ¹₃ of her total number of balls to Opal. In round 3, Opal lost ¹₃ of her total number of balls to Risa. In the end, Risa, Gwen and Opal had 110, 72 and 64 balls respectively. How many balls did Gwen have at first?

	Risa	Gwen	Opal
		69
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	108
Before 2	78	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	96
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Risa, Gwen and Opal in the end	110	72	64

Working backwards.

2 p = 64

1 p = 64 ÷ 2 = 32

At the start of Round 3:
Number of balls that Opal had
= 3 p
= 3 x 32
= 96

2 u = 72

1 u = 72 ÷ 2 = 36

At the start of Round 2:
Number of balls that Gwen had
= 3 u
= 3 x 36
= 108

At the start of Round 2:
Number of balls that Risa had
= 110 - 32
= 78

2 b = 78

1 b = 78 ÷ 2 = 39

At the start of Round 1:
Number of balls that Gwen had
= 108 - 1 b
= 108 - 39
= 69

Answer(s): 69