George, Elijah and Mark had a total of 93 buttons. The ratio of Elijah's buttons to Mark's buttons was 8 : 9 at first. George and Elijah each gave away ¹₂ of their buttons. Given that the three boys had 69 buttons left, how many buttons did George have in the end?

	George	Elijah	Mark	Total
Comparing Elijah and Mark at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	93
Change	- 1 p	-1x4 = - 4 u		- 24
After	1 p	1x4 = 4 u	9 u	69

Total number of buttons that George and Elijah gave away
= 93 - 69
= 24

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 8 and 2 is 8.

1 p + 4 u = 93 - 69
1 p + 4 u = 24
1 p = 24 - 4 u --- (1)

1 p + 4 u + 9 u = 69
1 p + 13 u = 69
1 p = 69 - 13 u --- (2)

(1) = (2)
24 - 4 u = 69 - 13 u
13 u - 4 u = 69 - 24
13 u - 4 u = 45
9 u = 45

1 u = 45 ÷ 9 = 5

Substitute 1 u = 5 into (1).
1 p = 24 - 4 u
1 p = 24 - 4 x 5
1 p = 24 - 20

1 p = 4

Number of buttons that George had in the end
= 1 p
= 4

Answer(s): 4