Archie, Justin and Riordan had a total of 148 buttons. The ratio of Justin's buttons to Riordan's buttons was 8 : 5 at first. Archie and Justin each gave away ¹₂ of their buttons. Given that the three boys had 99 buttons left, how many buttons did Archie have in the end?

	Archie	Justin	Riordan	Total
Comparing Justin and Riordan at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	148
Change	- 1 p	-1x4 = - 4 u		- 49
After	1 p	1x4 = 4 u	5 u	99

Total number of buttons that Archie and Justin gave away
= 148 - 99
= 49

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

1 p + 4 u = 148 - 99
1 p + 4 u = 49
1 p = 49 - 4 u --- (1)

1 p + 4 u + 5 u = 99
1 p + 9 u = 99
1 p = 99 - 9 u --- (2)

(1) = (2)
49 - 4 u = 99 - 9 u
9 u - 4 u = 99 - 49
9 u - 4 u = 50
5 u = 50

1 u = 50 ÷ 5 = 10

Substitute 1 u = 10 into (1).
1 p = 49 - 4 u
1 p = 49 - 4 x 10
1 p = 49 - 40

1 p = 9

Number of buttons that Archie had in the end
= 1 p
= 9

Answer(s): 9