Zoe, Betty and Vanessa each had some marbles and decided to play a game with their marbles. In round 1, Zoe lost ¹₂ of her marbles to Betty. In round 2, Betty lost ¹₂ of her total number of marbles to Vanessa. In round 3, Vanessa lost ¹₂ of her total number of marbles to Zoe. In the end, Zoe, Betty and Vanessa had 135, 84 and 54 marbles respectively. How many marbles did Betty have at first?

	Zoe	Betty	Vanessa
Before  1	2 b	87
Change 1	- 1 b	+ 1 b
After 1	1 b	168
Before 2	81	2 u
Change 2		- 1 u	+1
After 2		1 u	108
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Zoe, Betty and Vanessa in the end	135	84	54

Working backwards.

1 p = 54

At the start of Round 3:
Number of marbles that Vanessa had
= 2 p
= 2 x 54 
= 108

At the start of Round 2:
Number of marbles that Zoe had
= 135 - 1 p
= 135 - 54
= 81

At the end of Round 3:
Number of marbles that Betty had = 84

1 u = 84

At the end of Round 1:
Number of marbles that Betty had
= 2 u
= 2 x 84 
= 168

1 b = 81 

At the end of Round 1:
Number of marbles that Zoe had = 81

At the start of Round 1:
Number of marbles that Betty had
= 168 - 1 b
= 168 - 81
= 87

Answer(s): 87