Simon, Cody and David had a total of 42 buttons. The ratio of Cody's buttons to David's buttons was 6 : 7 at first. Simon and Cody each gave away ¹₂ of their buttons. Given that the three boys had 28 buttons left, how many buttons did Simon have in the end?

	Simon	Cody	David	Total
Comparing Cody and David at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	42
Change	- 1 p	-1x3 = - 3 u		- 14
After	1 p	1x3 = 3 u	7 u	28

Total number of buttons that Simon and Cody gave away
= 42 - 28
= 14

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

1 p + 3 u = 42 - 28
1 p + 3 u = 14
1 p = 14 - 3 u --- (1)

1 p + 3 u + 7 u = 28
1 p + 10 u = 28
1 p = 28 - 10 u --- (2)

(1) = (2)
14 - 3 u = 28 - 10 u
10 u - 3 u = 28 - 14
10 u - 3 u = 14
7 u = 14

1 u = 14 ÷ 7 = 2

Substitute 1 u = 2 into (1).
1 p = 14 - 3 u
1 p = 14 - 3 x 2
1 p = 14 - 6

1 p = 8

Number of buttons that Simon had in the end
= 1 p
= 8

Answer(s): 8