Betty, Cathy and Gwen were playing with beads. Betty started with 40 beads and in the first game, she won ⁴₅ more beads from Cathy. In the second game, Betty lost ⁵₈ of her original number of beads to Gwen. Cathy lost as many beads as Betty to Gwen. In the final game, Betty won ¹₂ of her original number of beads from Cathy. All of them ended up with the same number of beads.

1. How many beads did Betty have in the end?
2. How many beads did Cathy have at first?

	Betty	Cathy	Gwen
Before	40
First game	+ 32	- 32
Second game	- 25		+ 25
		- 25	+ 25
Final game	+ 20	- 20
After	1 u	1 u	1 u

(a)
Number of beads that Betty won from Cathy
= ⁴₅ x 40
= 32

Number of beads that Betty lost to Gwen
= ⁵₈ x 40
= 25

Number of beads that Betty won from Cathy
= ¹₂ x 40
= 20

Number of beads that Betty had in the end
= 40 + 32 - 25 + 20
= 67

(b)
Number of beads that Cathy had at first
= 1 u + 20 + 25 + 32
= 67 + 20 + 25 + 32
= 144 

Answer(s): (a) 67 ; (b) 144