Penelope, Kimberly and Zara each had some balls and decided to play a game with their balls. In round 1, Penelope lost ¹₂ of her balls to Kimberly. In round 2, Kimberly lost ¹₂ of her total number of balls to Zara. In round 3, Zara lost ¹₂ of her total number of balls to Penelope. In the end, Penelope, Kimberly and Zara had 112, 102 and 52 balls respectively. How many balls did Kimberly have at first?

	Penelope	Kimberly	Zara
Before  1	2 b	144
Change 1	- 1 b	+ 1 b
After 1	1 b	204
Before 2	60	2 u
Change 2		- 1 u	+1
After 2		1 u	104
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Penelope, Kimberly and Zara in the end	112	102	52

Working backwards.

1 p = 52

At the start of Round 3:
Number of balls that Zara had
= 2 p
= 2 x 52 
= 104

At the start of Round 2:
Number of balls that Penelope had
= 112 - 1 p
= 112 - 52
= 60

At the end of Round 3:
Number of balls that Kimberly had = 102

1 u = 102

At the end of Round 1:
Number of balls that Kimberly had
= 2 u
= 2 x 102 
= 204

1 b = 60 

At the end of Round 1:
Number of balls that Penelope had = 60

At the start of Round 1:
Number of balls that Kimberly had
= 204 - 1 b
= 204 - 60
= 144

Answer(s): 144