Sam had ³₄ as many marbles as Zeph. Zeph had ⁴₇ as many marbles as Xuan. After Sam and Xuan gave 22 marbles to Zeph, Zeph had thrice as many marbles as Sam. Xuan then had the same number of marbles as Zeph.

1. What was the ratio of the number of marbles Sam had to the number of marbles Xuan had in the beginning? Answer in simplest form.
2. How many marbles did Zeph have in the end?

Sam	Zeph	Xuan
3	4
	4	7
3	4	7

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

Ratio of the number of marbles that Sam had to the number of marbles that Xuan had in the beginning = 3 : 7

	Sam	Zeph	Xuan	Total
Before	3x1 = 3 u	4x1 = 4 u	7x1 = 7 u	14x1 = 14 u
Change	- 22	+ 44	- 22
After	1x2 = 2 u	3x2 = 6 u	3x2 = 6 u	7x2 = 14 u

The total number of marbles at first and in the end remains unchanged. Make the total number of marbles the same. LCM of 14 and 7 is 14.

Number of marbles that Zeph received from Sam and Xuan
= 2 x 22
= 44

Number of marbles that Zeph received from Sam and Xuan
= 2 x 1 u
= 2 u

2 u = 22

1 u = 22 ÷ 2 = 11

Number of marbles that Ashley had in the end
= 6 u
= 6 x 11
= 66