Luke had ¹₂ as many beads as Mark. Mark had ²₃ as many beads as Zara. After Luke and Zara gave 22 beads to Mark, Mark had four times as many beads as Luke. Zara then had the same number of beads as Mark.

1. What was the ratio of the number of beads Zara had to the number of beads Luke had in the beginning? Answer in simplest form.
2. How many beads did Mark have in the end?

Luke	Mark	Zara
1	2
	2	3
1	2	3

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

Ratio of the number of beads that Zara had to the number of beads that Luke had in the beginning = 3 : 1

	Luke	Mark	Zara	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 22	+ 44	- 22
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 Mark received from Luke and Zara
= 2 x 22
= 44

Number of beads that Mark received from Luke and Zara
= 2 x 1 u
= 2 u

2 u = 22

1 u = 22 ÷ 2 = 11

Number of beads that Ashley had in the end
= 8 u
= 8 x 11
= 88