Xylia, Kimberly and Abi each had some beads and decided to play a game with their beads. In round 1, Xylia lost ¹₃ of her beads to Kimberly. In round 2, Kimberly lost ¹₃ of her total number of beads to Abi. In round 3, Abi lost ¹₃ of her total number of beads to Xylia. In the end, Xylia, Kimberly and Abi had 140, 100 and 76 beads respectively. How many beads did Kimberly have at first?

	Xylia	Kimberly	Abi
		99
Before 1	3 b
Change 1	- 1 b	+ 1 b
After 1	2 b	150
Before 2	102	3 u
Change 2		- 1 u	+ 1 u
After 2		2 u	114
Before 3			3 p
Change 3	+ 1 p		- 1 p
After 3			2 p
Comparing Xylia, Kimberly and Abi in the end	140	100	76

Working backwards.

2 p = 76

1 p = 76 ÷ 2 = 38

At the start of Round 3:
Number of beads that Abi had
= 3 p
= 3 x 38
= 114

2 u = 100

1 u = 100 ÷ 2 = 50

At the start of Round 2:
Number of beads that Kimberly had
= 3 u
= 3 x 50
= 150

At the start of Round 2:
Number of beads that Xylia had
= 140 - 38
= 102

2 b = 102

1 b = 102 ÷ 2 = 51

At the start of Round 1:
Number of beads that Kimberly had
= 150 - 1 b
= 150 - 51
= 99

Answer(s): 99