Oliver, Ethan and Owen had a total of 112 stickers. The ratio of Ethan's stickers to Owen's stickers was 8 : 7 at first. Oliver and Ethan each gave away ¹₂ of their stickers. Given that the three boys had 77 stickers left, how many stickers did Oliver have in the end?

	Oliver	Ethan	Owen	Total
Comparing Ethan and Owen at first		8 u	7 u
Before	2 p	2x4 = 8 u	7 u	112
Change	- 1 p	-1x4 = - 4 u		- 35
After	1 p	1x4 = 4 u	7 u	77

Total number of stickers that Oliver and Ethan gave away
= 112 - 77
= 35

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

1 p + 4 u = 112 - 77
1 p + 4 u = 35
1 p = 35 - 4 u --- (1)

1 p + 4 u + 7 u = 77
1 p + 11 u = 77
1 p = 77 - 11 u --- (2)

(1) = (2)
35 - 4 u = 77 - 11 u
11 u - 4 u = 77 - 35
11 u - 4 u = 42
7 u = 42

1 u = 42 ÷ 7 = 6

Substitute 1 u = 6 into (1).
1 p = 35 - 4 u
1 p = 35 - 4 x 6
1 p = 35 - 24

1 p = 11

Number of stickers that Oliver had in the end
= 1 p
= 11

Answer(s): 11