Raeann, Zoe and Joelle each had some balls and decided to play a game with their balls. In round 1, Raeann lost ¹₃ of her balls to Zoe. In round 2, Zoe lost ¹₃ of her total number of balls to Joelle. In round 3, Joelle lost ¹₃ of her total number of balls to Raeann. In the end, Raeann, Zoe and Joelle had 120, 100 and 72 balls respectively. How many balls did Zoe have at first?

	Raeann	Zoe	Joelle
		108
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	150
Before 2	84	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	108
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Raeann, Zoe and Joelle in the end	120	100	72

Working backwards.

2 p = 72

1 p = 72 ÷ 2 = 36

At the start of Round 3:
Number of balls that Joelle had
= 3 p
= 3 x 36
= 108

2 u = 100

1 u = 100 ÷ 2 = 50

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

At the start of Round 2:
Number of balls that Raeann had
= 120 - 36
= 84

2 b = 84

1 b = 84 ÷ 2 = 42

At the start of Round 1:
Number of balls that Zoe had
= 150 - 1 b
= 150 - 42
= 108

Answer(s): 108