Jeremy had ¹₂ as many beads as Ethan. Ethan had ²₃ as many beads as Sarah. After Jeremy and Sarah gave 24 beads to Ethan, Ethan had four times as many beads as Jeremy. Sarah then had the same number of beads as Ethan.

1. What was the ratio of the number of beads Jeremy had to the number of beads Sarah had in the beginning? Answer in simplest form.
2. How many beads did Ethan have in the end?

Jeremy	Ethan	Sarah
1	2
	2	3
1	2	3

(a)
The number of beads that Ethan is repeated.  

Ratio of the number of beads that Jeremy had to the number of beads that Sarah had in the beginning = 1 : 3

	Jeremy	Ethan	Sarah	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 24	+ 48	- 24
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

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

Number of beads that Ethan received from Jeremy and Sarah
= 2 x 24
= 48

Number of beads that Ethan received from Jeremy and Sarah
= 2 x 1 u
= 2 u

2 u = 24

1 u = 24 ÷ 2 = 12

Number of beads that Ashley had in the end
= 8 u
= 8 x 12
= 96