Michael, Gabriel and Seth had a total of 157 stickers. The ratio of Gabriel's stickers to Seth's stickers was 8 : 7 at first. Michael and Gabriel each gave away ¹₂ of their stickers. Given that the three boys had 110 stickers left, how many stickers did Michael have at first?

	Michael	Gabriel	Seth	Total
Comparing Gabriel and Seth at first		8 u	7 u
Before	2 p	2x4 = 8 u	7 u	157
Change	- 1 p	-1x4 = - 4 u		- 47
After	1 p	1x4 = 4 u	7 u	110

Total number of stickers that Michael and Gabriel gave away
= 157 - 110
= 47

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

1 p + 4 u = 157 - 110
1 p + 4 u = 47
1 p = 47 - 4 u --- (1)

1 p + 4 u + 7 u = 110
1 p + 11 u = 110
1 p = 110 - 11 u --- (2)

(1) = (2)
47 - 4 u = 110 - 11 u
11 u - 4 u = 110 - 47
11 u - 4 u = 63
7 u = 63

1 u = 63 ÷ 7 = 9

Substitute 1 u = 9 into (1).
1 p = 47 - 4 u
1 p = 47 - 4 x 9
1 p = 47 - 36

1 p = 11

Number of stickers that Michael had at first
= 2 p
= 2 x 11
= 22

Answer(s): 22