Kimberly, Roshel and Gwen were playing with beads. Kimberly started with 63 beads and in the first game, she won ²₃ more beads from Roshel. In the second game, Kimberly lost ³₇ of her original number of beads to Gwen. Roshel lost as many beads as Kimberly to Gwen. In the final game, Kimberly won ¹₃ of her original number of beads from Roshel. All of them ended up with the same number of beads.

1. How many beads did Kimberly have in the end?
2. How many beads did Roshel have at first?

	Kimberly	Roshel	Gwen
Before	63
First game	+ 42	- 42
Second game	- 27		+ 27
		- 27	+ 27
Final game	+ 21	- 21
After	1 u	1 u	1 u

(a)
Number of beads that Kimberly won from Roshel
= ²₃ x 63
= 42

Number of beads that Kimberly lost to Gwen
= ³₇ x 63
= 27

Number of beads that Kimberly won from Roshel
= ¹₃ x 63
= 21

Number of beads that Kimberly had in the end
= 63 + 42 - 27 + 21
= 99

(b)
Number of beads that Roshel had at first
= 1 u + 21 + 27 + 42
= 99 + 21 + 27 + 42
= 189 

Answer(s): (a) 99 ; (b) 189