Irene, Elyse and Lucy each had some marbles and decided to play a game with their marbles. In round 1, Irene lost ¹₃ of her marbles to Elyse. In round 2, Elyse lost ¹₃ of her total number of marbles to Lucy. In round 3, Lucy lost ¹₃ of her total number of marbles to Irene. In the end, Irene, Elyse and Lucy had 137, 76 and 62 marbles respectively. How many marbles did Elyse have at first?

	Irene	Elyse	Lucy
		61
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	114
Before 2	106	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	93
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Irene, Elyse and Lucy in the end	137	76	62

Working backwards.

2 p = 62

1 p = 62 ÷ 2 = 31

At the start of Round 3:
Number of marbles that Lucy had
= 3 p
= 3 x 31
= 93

2 u = 76

1 u = 76 ÷ 2 = 38

At the start of Round 2:
Number of marbles that Elyse had
= 3 u
= 3 x 38
= 114

At the start of Round 2:
Number of marbles that Irene had
= 137 - 31
= 106

2 b = 106

1 b = 106 ÷ 2 = 53

At the start of Round 1:
Number of marbles that Elyse had
= 114 - 1 b
= 114 - 53
= 61

Answer(s): 61