Asher, Elijah and Harry had a total of 108 buttons. The ratio of Elijah's buttons to Harry's buttons was 4 : 9 at first. Asher and Elijah each gave away ¹₂ of their buttons. Given that the three boys had 90 buttons left, how many buttons did Asher have in the end?

	Asher	Elijah	Harry	Total
Comparing Elijah and Harry at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	108
Change	- 1 p	-1x2 = - 2 u		- 18
After	1 p	1x2 = 2 u	9 u	90

Total number of buttons that Asher and Elijah gave away
= 108 - 90
= 18

The number of buttons that Elijah had at first is repeated.  Make the number of buttons that Elijah had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 108 - 90
1 p + 2 u = 18
1 p = 18 - 2 u --- (1)

1 p + 2 u + 9 u = 90
1 p + 11 u = 90
1 p = 90 - 11 u --- (2)

(1) = (2)
18 - 2 u = 90 - 11 u
11 u - 2 u = 90 - 18
11 u - 2 u = 72
9 u = 72

1 u = 72 ÷ 9 = 8

Substitute 1 u = 8 into (1).
1 p = 18 - 2 u
1 p = 18 - 2 x 8
1 p = 18 - 16

1 p = 2

Number of buttons that Asher had in the end
= 1 p
= 2

Answer(s): 2