Zane, Perry and Flynn had a total of 137 buttons. The ratio of Perry's buttons to Flynn's buttons was 4 : 7 at first. Zane and Perry each gave away ¹₂ of their buttons. Given that the three boys had 107 buttons left, how many buttons did Zane have at first?

	Zane	Perry	Flynn	Total
Comparing Perry and Flynn at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	137
Change	- 1 p	-1x2 = - 2 u		- 30
After	1 p	1x2 = 2 u	7 u	107

Total number of buttons that Zane and Perry gave away
= 137 - 107
= 30

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

1 p + 2 u = 137 - 107
1 p + 2 u = 30
1 p = 30 - 2 u --- (1)

1 p + 2 u + 7 u = 107
1 p + 9 u = 107
1 p = 107 - 9 u --- (2)

(1) = (2)
30 - 2 u = 107 - 9 u
9 u - 2 u = 107 - 30
9 u - 2 u = 77
7 u = 77

1 u = 77 ÷ 7 = 11

Substitute 1 u = 11 into (1).
1 p = 30 - 2 u
1 p = 30 - 2 x 11
1 p = 30 - 22

1 p = 8

Number of buttons that Zane had at first
= 2 p
= 2 x 8
= 16

Answer(s): 16