Howard had ³₄ as many buttons as Harry. Harry had ⁴₇ as many buttons as Xuan. After Howard and Xuan gave 22 buttons to Harry, Harry had thrice as many buttons as Howard. Xuan then had the same number of buttons as Harry.

1. What was the ratio of the number of buttons Xuan had to the number of buttons Howard had in the beginning? Answer in simplest form.
2. How many buttons did Harry have in the end?

Howard	Harry	Xuan
3	4
	4	7
3	4	7

(a)
The number of buttons that Harry is repeated.  

Ratio of the number of buttons that Xuan had to the number of buttons that Howard had in the beginning = 7 : 3

	Howard	Harry	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 buttons at first and in the end remains unchanged. Make the total number of buttons the same. LCM of 14 and 7 is 14.

Number of buttons that Harry received from Howard and Xuan
= 2 x 22
= 44

Number of buttons that Harry received from Howard and Xuan
= 2 x 1 u
= 2 u

2 u = 22

1 u = 22 ÷ 2 = 11

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