Gabby, Nicole and Raeann each had some balls and decided to play a game with their balls. In round 1, Gabby lost ¹₃ of her balls to Nicole. In round 2, Nicole lost ¹₃ of her total number of balls to Raeann. In round 3, Raeann lost ¹₃ of her total number of balls to Gabby. In the end, Gabby, Nicole and Raeann had 127, 86 and 78 balls respectively. How many balls did Nicole have at first?

	Gabby	Nicole	Raeann
		85
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	129
Before 2	88	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	117
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Gabby, Nicole and Raeann in the end	127	86	78

Working backwards.

2 p = 78

1 p = 78 ÷ 2 = 39

At the start of Round 3:
Number of balls that Raeann had
= 3 p
= 3 x 39
= 117

2 u = 86

1 u = 86 ÷ 2 = 43

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

At the start of Round 2:
Number of balls that Gabby had
= 127 - 39
= 88

2 b = 88

1 b = 88 ÷ 2 = 44

At the start of Round 1:
Number of balls that Nicole had
= 129 - 1 b
= 129 - 44
= 85

Answer(s): 85