Lucy, Zara and Gillian each had some marbles and decided to play a game with their marbles. In round 1, Lucy lost ¹₃ of her marbles to Zara. In round 2, Zara lost ¹₃ of her total number of marbles to Gillian. In round 3, Gillian lost ¹₃ of her total number of marbles to Lucy. In the end, Lucy, Zara and Gillian had 120, 98 and 68 marbles respectively. How many marbles did Zara have at first?

	Lucy	Zara	Gillian
		104
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	147
Before 2	86	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	102
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Lucy, Zara and Gillian in the end	120	98	68

Working backwards.

2 p = 68

1 p = 68 ÷ 2 = 34

At the start of Round 3:
Number of marbles that Gillian had
= 3 p
= 3 x 34
= 102

2 u = 98

1 u = 98 ÷ 2 = 49

At the start of Round 2:
Number of marbles that Zara had
= 3 u
= 3 x 49
= 147

At the start of Round 2:
Number of marbles that Lucy had
= 120 - 34
= 86

2 b = 86

1 b = 86 ÷ 2 = 43

At the start of Round 1:
Number of marbles that Zara had
= 147 - 1 b
= 147 - 43
= 104

Answer(s): 104