Zane had ¹₂ as many marbles as Ivan. Ivan had ²₃ as many marbles as Nicole. After Zane and Nicole gave 8 marbles to Ivan, Ivan had four times as many marbles as Zane. Nicole then had the same number of marbles as Ivan.

1. What was the ratio of the number of marbles Zane had to the number of marbles Nicole had in the beginning? Answer in simplest form.
2. How many marbles did Ivan have in the end?

Zane	Ivan	Nicole
1	2
	2	3
1	2	3

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

Ratio of the number of marbles that Zane had to the number of marbles that Nicole had in the beginning = 1 : 3

	Zane	Ivan	Nicole	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 8	+ 16	- 8
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

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

Number of marbles that Ivan received from Zane and Nicole
= 2 x 8
= 16

Number of marbles that Ivan received from Zane and Nicole
= 2 x 1 u
= 2 u

2 u = 8

1 u = 8 ÷ 2 = 4

Number of marbles that Ashley had in the end
= 8 u
= 8 x 4
= 32