Japheth, Luis and Simon had a total of 141 buttons. The ratio of Luis's buttons to Simon's buttons was 8 : 9 at first. Japheth and Luis each gave away ¹₂ of their buttons. Given that the three boys had 102 buttons left, how many buttons did Japheth have at first?

	Japheth	Luis	Simon	Total
Comparing Luis and Simon at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	141
Change	- 1 p	-1x4 = - 4 u		- 39
After	1 p	1x4 = 4 u	9 u	102

Total number of buttons that Japheth and Luis gave away
= 141 - 102
= 39

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

1 p + 4 u = 141 - 102
1 p + 4 u = 39
1 p = 39 - 4 u --- (1)

1 p + 4 u + 9 u = 102
1 p + 13 u = 102
1 p = 102 - 13 u --- (2)

(1) = (2)
39 - 4 u = 102 - 13 u
13 u - 4 u = 102 - 39
13 u - 4 u = 63
9 u = 63

1 u = 63 ÷ 9 = 7

Substitute 1 u = 7 into (1).
1 p = 39 - 4 u
1 p = 39 - 4 x 7
1 p = 39 - 28

1 p = 11

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

Answer(s): 22