Lucy, Penelope and Tina each had some marbles and decided to play a game with their marbles. In round 1, Lucy lost ¹₃ of her marbles to Penelope. In round 2, Penelope lost ¹₃ of her total number of marbles to Tina. In round 3, Tina lost ¹₃ of her total number of marbles to Lucy. In the end, Lucy, Penelope and Tina had 121, 82 and 78 marbles respectively. How many marbles did Penelope have at first?

	Lucy	Penelope	Tina
		82
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	123
Before 2	82	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	117
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Lucy, Penelope and Tina in the end	121	82	78

Working backwards.

2 p = 78

1 p = 78 ÷ 2 = 39

At the start of Round 3:
Number of marbles that Tina had
= 3 p
= 3 x 39
= 117

2 u = 82

1 u = 82 ÷ 2 = 41

At the start of Round 2:
Number of marbles that Penelope had
= 3 u
= 3 x 41
= 123

At the start of Round 2:
Number of marbles that Lucy had
= 121 - 39
= 82

2 b = 82

1 b = 82 ÷ 2 = 41

At the start of Round 1:
Number of marbles that Penelope had
= 123 - 1 b
= 123 - 41
= 82

Answer(s): 82