Valen had ¹₂ as many stickers as Ivan. Ivan had ²₃ as many stickers as Cathy. After Valen and Cathy gave 10 stickers to Ivan, Ivan had four times as many stickers as Valen. Cathy then had the same number of stickers as Ivan.

1. What was the ratio of the number of stickers Valen had to the number of stickers Cathy had in the beginning? Answer in simplest form.
2. How many stickers did Ivan have in the end?

Valen	Ivan	Cathy
1	2
	2	3
1	2	3

(a)
The number of stickers that Ivan is repeated.  

Ratio of the number of stickers that Valen had to the number of stickers that Cathy had in the beginning = 1 : 3

	Valen	Ivan	Cathy	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 10	+ 20	- 10
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

The total number of stickers at first and in the end remains unchanged. Make the total number of stickers the same. LCM of 6 and 9 is 18.

Number of stickers that Ivan received from Valen and Cathy
= 2 x 10
= 20

Number of stickers that Ivan received from Valen and Cathy
= 2 x 1 u
= 2 u

2 u = 10

1 u = 10 ÷ 2 = 5

Number of stickers that Ashley had in the end
= 8 u
= 8 x 5
= 40