Bobby, Glen and Ivan had a total of 72 buttons. The ratio of Glen's buttons to Ivan's buttons was 8 : 7 at first. Bobby and Glen each gave away ¹₂ of their buttons. Given that the three boys had 50 buttons left, how many buttons did Bobby have in the end?

	Bobby	Glen	Ivan	Total
Comparing Glen and Ivan at first		8 u	7 u
Before	2 p	2x4 = 8 u	7 u	72
Change	- 1 p	-1x4 = - 4 u		- 22
After	1 p	1x4 = 4 u	7 u	50

Total number of buttons that Bobby and Glen gave away
= 72 - 50
= 22

The number of buttons that Glen had at first is repeated.  Make the number of buttons that Glen had at first the same.  LCM of 8 and 2 is 8.

1 p + 4 u = 72 - 50
1 p + 4 u = 22
1 p = 22 - 4 u --- (1)

1 p + 4 u + 7 u = 50
1 p + 11 u = 50
1 p = 50 - 11 u --- (2)

(1) = (2)
22 - 4 u = 50 - 11 u
11 u - 4 u = 50 - 22
11 u - 4 u = 28
7 u = 28

1 u = 28 ÷ 7 = 4

Substitute 1 u = 4 into (1).
1 p = 22 - 4 u
1 p = 22 - 4 x 4
1 p = 22 - 16

1 p = 6

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

Answer(s): 6