Ryan, Cole and Michael had a total of 74 buttons. The ratio of Cole's buttons to Michael's buttons was 8 : 5 at first. Ryan and Cole each gave away ¹₂ of their buttons. Given that the three boys had 47 buttons left, how many buttons did Ryan have at first?

	Ryan	Cole	Michael	Total
Comparing Cole and Michael at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	74
Change	- 1 p	-1x4 = - 4 u		- 27
After	1 p	1x4 = 4 u	5 u	47

Total number of buttons that Ryan and Cole gave away
= 74 - 47
= 27

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

1 p + 4 u = 74 - 47
1 p + 4 u = 27
1 p = 27 - 4 u --- (1)

1 p + 4 u + 5 u = 47
1 p + 9 u = 47
1 p = 47 - 9 u --- (2)

(1) = (2)
27 - 4 u = 47 - 9 u
9 u - 4 u = 47 - 27
9 u - 4 u = 20
5 u = 20

1 u = 20 ÷ 5 = 4

Substitute 1 u = 4 into (1).
1 p = 27 - 4 u
1 p = 27 - 4 x 4
1 p = 27 - 16

1 p = 11

Number of buttons that Ryan had at first
= 2 p
= 2 x 11
= 22

Answer(s): 22