Seth, Michael and Ahmad had a total of 28 buttons. The ratio of Michael's buttons to Ahmad's buttons was 6 : 5 at first. Seth and Michael each gave away ¹₂ of their buttons. Given that the three boys had 19 buttons left, how many buttons did Seth have in the end?

	Seth	Michael	Ahmad	Total
Comparing Michael and Ahmad at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	28
Change	- 1 p	-1x3 = - 3 u		- 9
After	1 p	1x3 = 3 u	5 u	19

Total number of buttons that Seth and Michael gave away
= 28 - 19
= 9

The number of buttons that Michael had at first is repeated.  Make the number of buttons that Michael had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 28 - 19
1 p + 3 u = 9
1 p = 9 - 3 u --- (1)

1 p + 3 u + 5 u = 19
1 p + 8 u = 19
1 p = 19 - 8 u --- (2)

(1) = (2)
9 - 3 u = 19 - 8 u
8 u - 3 u = 19 - 9
8 u - 3 u = 10
5 u = 10

1 u = 10 ÷ 5 = 2

Substitute 1 u = 2 into (1).
1 p = 9 - 3 u
1 p = 9 - 3 x 2
1 p = 9 - 6

1 p = 3

Number of buttons that Seth had in the end
= 1 p
= 3

Answer(s): 3