Usha, Irene and Nicole each had some marbles and decided to play a game with their marbles. In round 1, Usha lost ¹₂ of her marbles to Irene. In round 2, Irene lost ¹₂ of her total number of marbles to Nicole. In round 3, Nicole lost ¹₂ of her total number of marbles to Usha. In the end, Usha, Irene and Nicole had 114, 81 and 54 marbles respectively. How many marbles did Irene have at first?

	Usha	Irene	Nicole
Before  1	2 b	102
Change 1	- 1 b	+ 1 b
After 1	1 b	162
Before 2	60	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 Usha, Irene and Nicole in the end	114	81	54

Working backwards.

1 p = 54

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

At the start of Round 2:
Number of marbles that Usha had
= 114 - 1 p
= 114 - 54
= 60

At the end of Round 3:
Number of marbles that Irene had = 81

1 u = 81

At the end of Round 1:
Number of marbles that Irene had
= 2 u
= 2 x 81 
= 162

1 b = 60 

At the end of Round 1:
Number of marbles that Usha had = 60

At the start of Round 1:
Number of marbles that Irene had
= 162 - 1 b
= 162 - 60
= 102

Answer(s): 102