John, Reggie and Albert had a total of 86 buttons. The ratio of Reggie's buttons to Albert's buttons was 6 : 5 at first. John and Reggie each gave away ¹₂ of their buttons. Given that the three boys had 58 buttons left, how many buttons did John have in the end?

	John	Reggie	Albert	Total
Comparing Reggie and Albert at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	86
Change	- 1 p	-1x3 = - 3 u		- 28
After	1 p	1x3 = 3 u	5 u	58

Total number of buttons that John and Reggie gave away
= 86 - 58
= 28

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

1 p + 3 u = 86 - 58
1 p + 3 u = 28
1 p = 28 - 3 u --- (1)

1 p + 3 u + 5 u = 58
1 p + 8 u = 58
1 p = 58 - 8 u --- (2)

(1) = (2)
28 - 3 u = 58 - 8 u
8 u - 3 u = 58 - 28
8 u - 3 u = 30
5 u = 30

1 u = 30 ÷ 5 = 6

Substitute 1 u = 6 into (1).
1 p = 28 - 3 u
1 p = 28 - 3 x 6
1 p = 28 - 18

1 p = 10

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

Answer(s): 10