Jack, Pierre and Sam had a total of 67 buttons. The ratio of Pierre's buttons to Sam's buttons was 6 : 5 at first. Jack and Pierre each gave away ¹₂ of their buttons. Given that the three boys had 46 buttons left, how many buttons did Jack have in the end?

	Jack	Pierre	Sam	Total
Comparing Pierre and Sam at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	67
Change	- 1 p	-1x3 = - 3 u		- 21
After	1 p	1x3 = 3 u	5 u	46

Total number of buttons that Jack and Pierre gave away
= 67 - 46
= 21

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

1 p + 3 u = 67 - 46
1 p + 3 u = 21
1 p = 21 - 3 u --- (1)

1 p + 3 u + 5 u = 46
1 p + 8 u = 46
1 p = 46 - 8 u --- (2)

(1) = (2)
21 - 3 u = 46 - 8 u
8 u - 3 u = 46 - 21
8 u - 3 u = 25
5 u = 25

1 u = 25 ÷ 5 = 5

Substitute 1 u = 5 into (1).
1 p = 21 - 3 u
1 p = 21 - 3 x 5
1 p = 21 - 15

1 p = 6

Number of buttons that Jack had in the end
= 1 p
= 6

Answer(s): 6