Michael, Tim and George had a total of 55 buttons. The ratio of Tim's buttons to George's buttons was 4 : 3 at first. Michael and Tim each gave away ¹₂ of their buttons. Given that the three boys had 38 buttons left, how many buttons did Michael have at first?

	Michael	Tim	George	Total
Comparing Tim and George at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	55
Change	- 1 p	-1x2 = - 2 u		- 17
After	1 p	1x2 = 2 u	3 u	38

Total number of buttons that Michael and Tim gave away
= 55 - 38
= 17

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

1 p + 2 u = 55 - 38
1 p + 2 u = 17
1 p = 17 - 2 u --- (1)

1 p + 2 u + 3 u = 38
1 p + 5 u = 38
1 p = 38 - 5 u --- (2)

(1) = (2)
17 - 2 u = 38 - 5 u
5 u - 2 u = 38 - 17
5 u - 2 u = 21
3 u = 21

1 u = 21 ÷ 3 = 7

Substitute 1 u = 7 into (1).
1 p = 17 - 2 u
1 p = 17 - 2 x 7
1 p = 17 - 14

1 p = 3

Number of buttons that Michael had at first
= 2 p
= 2 x 3
= 6

Answer(s): 6