Julian, Ben and Vincent had a total of 122 buttons. The ratio of Ben's buttons to Vincent's buttons was 2 : 9 at first. Julian and Ben each gave away ¹₂ of their buttons. Given that the three boys had 106 buttons left, how many buttons did Julian have at first?

	Julian	Ben	Vincent	Total
Comparing Ben and Vincent at first		2 u	9 u
Before	2 p	2x1 = 2 u	9 u	122
Change	- 1 p	-1x1 = - 1 u		- 16
After	1 p	1x1 = 1 u	9 u	106

Total number of buttons that Julian and Ben gave away
= 122 - 106
= 16

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

1 p + 1 u = 122 - 106
1 p + 1 u = 16
1 p = 16 - 1 u --- (1)

1 p + 1 u + 9 u = 106
1 p + 10 u = 106
1 p = 106 - 10 u --- (2)

(1) = (2)
16 - 1 u = 106 - 10 u
10 u - 1 u = 106 - 16
10 u - 1 u = 90
9 u = 90

1 u = 90 ÷ 9 = 10

Substitute 1 u = 10 into (1).
1 p = 16 - 1 u
1 p = 16 - 1 x 10
1 p = 16 - 10

1 p = 6

Number of buttons that Julian had at first
= 2 p
= 2 x 6
= 12

Answer(s): 12