Penelope, Rachel and Xylia were playing with beads. Penelope started with 24 beads and in the first game, she won ¹₃ more beads from Rachel. In the second game, Penelope lost ¹₈ of her original number of beads to Xylia. Rachel lost as many beads as Penelope to Xylia. In the final game, Penelope won ¹₂ of her original number of beads from Rachel. All of them ended up with the same number of beads.

1. How many beads did Penelope have in the end?
2. How many beads did Rachel have at first?

	Penelope	Rachel	Xylia
Before	24
First game	+ 8	- 8
Second game	- 3		+ 3
		- 3	+ 3
Final game	+ 12	- 12
After	1 u	1 u	1 u

(a)
Number of beads that Penelope won from Rachel
= ¹₃ x 24
= 8

Number of beads that Penelope lost to Xylia
= ¹₈ x 24
= 3

Number of beads that Penelope won from Rachel
= ¹₂ x 24
= 12

Number of beads that Penelope had in the end
= 24 + 8 - 3 + 12
= 41

(b)
Number of beads that Rachel had at first
= 1 u + 12 + 3 + 8
= 41 + 12 + 3 + 8
= 64 

Answer(s): (a) 41 ; (b) 64