Julian, Asher and Harry had a total of 125 buttons. The ratio of Asher's buttons to Harry's buttons was 6 : 5 at first. Julian and Asher each gave away ¹₂ of their buttons. Given that the three boys had 90 buttons left, how many buttons did Julian have in the end?

	Julian	Asher	Harry	Total
Comparing Asher and Harry at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	125
Change	- 1 p	-1x3 = - 3 u		- 35
After	1 p	1x3 = 3 u	5 u	90

Total number of buttons that Julian and Asher gave away
= 125 - 90
= 35

The number of buttons that Asher had at first is repeated.  Make the number of buttons that Asher had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 125 - 90
1 p + 3 u = 35
1 p = 35 - 3 u --- (1)

1 p + 3 u + 5 u = 90
1 p + 8 u = 90
1 p = 90 - 8 u --- (2)

(1) = (2)
35 - 3 u = 90 - 8 u
8 u - 3 u = 90 - 35
8 u - 3 u = 55
5 u = 55

1 u = 55 ÷ 5 = 11

Substitute 1 u = 11 into (1).
1 p = 35 - 3 u
1 p = 35 - 3 x 11
1 p = 35 - 33

1 p = 2

Number of buttons that Julian had in the end
= 1 p
= 2

Answer(s): 2