Hazel, Dana and Yen each had some marbles and decided to play a game with their marbles. In round 1, Hazel lost ¹₂ of her marbles to Dana. In round 2, Dana lost ¹₂ of her total number of marbles to Yen. In round 3, Yen lost ¹₂ of her total number of marbles to Hazel. In the end, Hazel, Dana and Yen had 111, 87 and 57 marbles respectively. How many marbles did Dana have at first?

	Hazel	Dana	Yen
Before  1	2 b	120
Change 1	- 1 b	+ 1 b
After 1	1 b	174
Before 2	54	2 u
Change 2		- 1 u	+1
After 2		1 u	114
Before 3			2 p
Change 3	+ 1 p		- 1 p
After 3			1 p
Comparing Hazel, Dana and Yen in the end	111	87	57

Working backwards.

1 p = 57

At the start of Round 3:
Number of marbles that Yen had
= 2 p
= 2 x 57 
= 114

At the start of Round 2:
Number of marbles that Hazel had
= 111 - 1 p
= 111 - 57
= 54

At the end of Round 3:
Number of marbles that Dana had = 87

1 u = 87

At the end of Round 1:
Number of marbles that Dana had
= 2 u
= 2 x 87 
= 174

1 b = 54 

At the end of Round 1:
Number of marbles that Hazel had = 54

At the start of Round 1:
Number of marbles that Dana had
= 174 - 1 b
= 174 - 54
= 120

Answer(s): 120