Andy, Japheth and Peter had a total of 176 buttons. The ratio of Japheth's buttons to Peter's buttons was 6 : 7 at first. Andy and Japheth each gave away ¹₂ of their buttons. Given that the three boys had 130 buttons left, how many buttons did Andy have in the end?

	Andy	Japheth	Peter	Total
Comparing Japheth and Peter at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	176
Change	- 1 p	-1x3 = - 3 u		- 46
After	1 p	1x3 = 3 u	7 u	130

Total number of buttons that Andy and Japheth gave away
= 176 - 130
= 46

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

1 p + 3 u = 176 - 130
1 p + 3 u = 46
1 p = 46 - 3 u --- (1)

1 p + 3 u + 7 u = 130
1 p + 10 u = 130
1 p = 130 - 10 u --- (2)

(1) = (2)
46 - 3 u = 130 - 10 u
10 u - 3 u = 130 - 46
10 u - 3 u = 84
7 u = 84

1 u = 84 ÷ 7 = 12

Substitute 1 u = 12 into (1).
1 p = 46 - 3 u
1 p = 46 - 3 x 12
1 p = 46 - 36

1 p = 10

Number of buttons that Andy had in the end
= 1 p
= 10

Answer(s): 10