Zane had ¹₂ as many erasers as Paul. Paul had ²₃ as many erasers as Shiyun. After Zane and Shiyun gave 8 erasers to Paul, Paul had four times as many erasers as Zane. Shiyun then had the same number of erasers as Paul.

1. What was the ratio of the number of erasers Shiyun had to the number of erasers Zane had in the beginning? Answer in simplest form.
2. How many erasers did Paul have in the end?

Zane	Paul	Shiyun
1	2
	2	3
1	2	3

(a)
The number of erasers that Paul is repeated.  

Ratio of the number of erasers that Shiyun had to the number of erasers that Zane had in the beginning = 3 : 1

	Zane	Paul	Shiyun	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 erasers at first and in the end remains unchanged. Make the total number of erasers the same. LCM of 6 and 9 is 18.

Number of erasers that Paul received from Zane and Shiyun
= 2 x 8
= 16

Number of erasers that Paul received from Zane and Shiyun
= 2 x 1 u
= 2 u

2 u = 8

1 u = 8 ÷ 2 = 4

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