Yen, Sarah and Emily each had some balls and decided to play a game with their balls. In round 1, Yen lost ¹₂ of her balls to Sarah. In round 2, Sarah lost ¹₂ of her total number of balls to Emily. In round 3, Emily lost ¹₂ of her total number of balls to Yen. In the end, Yen, Sarah and Emily had 137, 105 and 59 balls respectively. How many balls did Sarah have at first?

	Yen	Sarah	Emily
Before  1	2 b	132
Change 1	- 1 b	+ 1 b
After 1	1 b	210
Before 2	78	2 u
Change 2		- 1 u	+1
After 2		1 u	118
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Yen, Sarah and Emily in the end	137	105	59

Working backwards.

1 p = 59

At the start of Round 3:
Number of balls that Emily had
= 2 p
= 2 x 59 
= 118

At the start of Round 2:
Number of balls that Yen had
= 137 - 1 p
= 137 - 59
= 78

At the end of Round 3:
Number of balls that Sarah had = 105

1 u = 105

At the end of Round 1:
Number of balls that Sarah had
= 2 u
= 2 x 105 
= 210

1 b = 78 

At the end of Round 1:
Number of balls that Yen had = 78

At the start of Round 1:
Number of balls that Sarah had
= 210 - 1 b
= 210 - 78
= 132

Answer(s): 132