Simon, Luis and George had a total of 82 buttons. The ratio of Luis's buttons to George's buttons was 4 : 7 at first. Simon and Luis each gave away ¹₂ of their buttons. Given that the three boys had 62 buttons left, how many buttons did Simon have in the end?

	Simon	Luis	George	Total
Comparing Luis and George at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	82
Change	- 1 p	-1x2 = - 2 u		- 20
After	1 p	1x2 = 2 u	7 u	62

Total number of buttons that Simon and Luis gave away
= 82 - 62
= 20

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

1 p + 2 u = 82 - 62
1 p + 2 u = 20
1 p = 20 - 2 u --- (1)

1 p + 2 u + 7 u = 62
1 p + 9 u = 62
1 p = 62 - 9 u --- (2)

(1) = (2)
20 - 2 u = 62 - 9 u
9 u - 2 u = 62 - 20
9 u - 2 u = 42
7 u = 42

1 u = 42 ÷ 7 = 6

Substitute 1 u = 6 into (1).
1 p = 20 - 2 u
1 p = 20 - 2 x 6
1 p = 20 - 12

1 p = 8

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

Answer(s): 8