Tim, Charlie and Ian had a total of 94 buttons. The ratio of Charlie's buttons to Ian's buttons was 4 : 9 at first. Tim and Charlie each gave away ¹₂ of their buttons. Given that the three boys had 74 buttons left, how many buttons did Tim have at first?

	Tim	Charlie	Ian	Total
Comparing Charlie and Ian at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	94
Change	- 1 p	-1x2 = - 2 u		- 20
After	1 p	1x2 = 2 u	9 u	74

Total number of buttons that Tim and Charlie gave away
= 94 - 74
= 20

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

1 p + 2 u = 94 - 74
1 p + 2 u = 20
1 p = 20 - 2 u --- (1)

1 p + 2 u + 9 u = 74
1 p + 11 u = 74
1 p = 74 - 11 u --- (2)

(1) = (2)
20 - 2 u = 74 - 11 u
11 u - 2 u = 74 - 20
11 u - 2 u = 54
9 u = 54

1 u = 54 ÷ 9 = 6

Substitute 1 u = 6 into (1).
1 p = 20 - 2 u
1 p = 20 - 2 x 6
1 p = 20 - 12

1 p = 8

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

Answer(s): 16