Ben, Eric and George had a total of 115 buttons. The ratio of Eric's buttons to George's buttons was 6 : 5 at first. Ben and Eric each gave away ¹₂ of their buttons. Given that the three boys had 80 buttons left, how many buttons did Ben have in the end?

	Ben	Eric	George	Total
Comparing Eric and George at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	115
Change	- 1 p	-1x3 = - 3 u		- 35
After	1 p	1x3 = 3 u	5 u	80

Total number of buttons that Ben and Eric gave away
= 115 - 80
= 35

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

1 p + 3 u = 115 - 80
1 p + 3 u = 35
1 p = 35 - 3 u --- (1)

1 p + 3 u + 5 u = 80
1 p + 8 u = 80
1 p = 80 - 8 u --- (2)

(1) = (2)
35 - 3 u = 80 - 8 u
8 u - 3 u = 80 - 35
8 u - 3 u = 45
5 u = 45

1 u = 45 ÷ 5 = 9

Substitute 1 u = 9 into (1).
1 p = 35 - 3 u
1 p = 35 - 3 x 9
1 p = 35 - 27

1 p = 8

Number of buttons that Ben had in the end
= 1 p
= 8

Answer(s): 8