Howard had ¹₂ as many buttons as Xavier. Xavier had ²₃ as many buttons as Emma. After Howard and Emma gave 18 buttons to Xavier, Xavier had four times as many buttons as Howard. Emma then had the same number of buttons as Xavier.

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

Howard	Xavier	Emma
1	2
	2	3
1	2	3

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

Ratio of the number of buttons that Howard had to the number of buttons that Emma had in the beginning = 1 : 3

	Howard	Xavier	Emma	Total
Before	1x3 = 3 u	2x3 = 6 u	3x3 = 9 u	6x3 = 18 u
Change	- 18	+ 36	- 18
After	1x2 = 2 u	4x2 = 8 u	4x2 = 8 u	9x2 = 18 u

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

Number of buttons that Xavier received from Howard and Emma
= 2 x 18
= 36

Number of buttons that Xavier received from Howard and Emma
= 2 x 1 u
= 2 u

2 u = 18

1 u = 18 ÷ 2 = 9

Number of buttons that Ashley had in the end
= 8 u
= 8 x 9
= 72