Gem, Opal and Jen each had some balls and decided to play a game with their balls. In round 1, Gem lost ¹₂ of her balls to Opal. In round 2, Opal lost ¹₂ of her total number of balls to Jen. In round 3, Jen lost ¹₂ of her total number of balls to Gem. In the end, Gem, Opal and Jen had 113, 105 and 62 balls respectively. How many balls did Opal have at first?

	Gem	Opal	Jen
Before  1	2 b	159
Change 1	- 1 b	+ 1 b
After 1	1 b	210
Before 2	51	2 u
Change 2		- 1 u	+1
After 2		1 u	124
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Gem, Opal and Jen in the end	113	105	62

Working backwards.

1 p = 62

At the start of Round 3:
Number of balls that Jen had
= 2 p
= 2 x 62 
= 124

At the start of Round 2:
Number of balls that Gem had
= 113 - 1 p
= 113 - 62
= 51

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

1 u = 105

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

1 b = 51 

At the end of Round 1:
Number of balls that Gem had = 51

At the start of Round 1:
Number of balls that Opal had
= 210 - 1 b
= 210 - 51
= 159

Answer(s): 159