Abi, Cathy and Hazel each had some beads and decided to play a game with their beads. In round 1, Abi lost ¹₃ of her beads to Cathy. In round 2, Cathy lost ¹₃ of her total number of beads to Hazel. In round 3, Hazel lost ¹₃ of her total number of beads to Abi. In the end, Abi, Cathy and Hazel had 110, 92 and 80 beads respectively. How many beads did Cathy have at first?

	Abi	Cathy	Hazel
		103
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	138
Before 2	70	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	120
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Abi, Cathy and Hazel in the end	110	92	80

Working backwards.

2 p = 80

1 p = 80 ÷ 2 = 40

At the start of Round 3:
Number of beads that Hazel had
= 3 p
= 3 x 40
= 120

2 u = 92

1 u = 92 ÷ 2 = 46

At the start of Round 2:
Number of beads that Cathy had
= 3 u
= 3 x 46
= 138

At the start of Round 2:
Number of beads that Abi had
= 110 - 40
= 70

2 b = 70

1 b = 70 ÷ 2 = 35

At the start of Round 1:
Number of beads that Cathy had
= 138 - 1 b
= 138 - 35
= 103

Answer(s): 103