Andy, Valen and Will had a total of 26 buttons. The ratio of Valen's buttons to Will's buttons was 2 : 3 at first. Andy and Valen each gave away ¹₂ of their buttons. Given that the three boys had 19 buttons left, how many buttons did Andy have at first?

	Andy	Valen	Will	Total
Comparing Valen and Will at first		2 u	3 u
Before	2 p	2x1 = 2 u	3 u	26
Change	- 1 p	-1x1 = - 1 u		- 7
After	1 p	1x1 = 1 u	3 u	19

Total number of buttons that Andy and Valen gave away
= 26 - 19
= 7

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

1 p + 1 u = 26 - 19
1 p + 1 u = 7
1 p = 7 - 1 u --- (1)

1 p + 1 u + 3 u = 19
1 p + 4 u = 19
1 p = 19 - 4 u --- (2)

(1) = (2)
7 - 1 u = 19 - 4 u
4 u - 1 u = 19 - 7
4 u - 1 u = 12
3 u = 12

1 u = 12 ÷ 3 = 4

Substitute 1 u = 4 into (1).
1 p = 7 - 1 u
1 p = 7 - 1 x 4
1 p = 7 - 4

1 p = 3

Number of buttons that Andy had at first
= 2 p
= 2 x 3
= 6

Answer(s): 6