Ethan, Caden and Rael had a total of 148 stickers. The ratio of Caden's stickers to Rael's stickers was 6 : 7 at first. Ethan and Caden each gave away ¹₂ of their stickers. Given that the three boys had 109 stickers left, how many stickers did Ethan have in the end?

	Ethan	Caden	Rael	Total
Comparing Caden and Rael at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	148
Change	- 1 p	-1x3 = - 3 u		- 39
After	1 p	1x3 = 3 u	7 u	109

Total number of stickers that Ethan and Caden gave away
= 148 - 109
= 39

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

1 p + 3 u = 148 - 109
1 p + 3 u = 39
1 p = 39 - 3 u --- (1)

1 p + 3 u + 7 u = 109
1 p + 10 u = 109
1 p = 109 - 10 u --- (2)

(1) = (2)
39 - 3 u = 109 - 10 u
10 u - 3 u = 109 - 39
10 u - 3 u = 70
7 u = 70

1 u = 70 ÷ 7 = 10

Substitute 1 u = 10 into (1).
1 p = 39 - 3 u
1 p = 39 - 3 x 10
1 p = 39 - 30

1 p = 9

Number of stickers that Ethan had in the end
= 1 p
= 9

Answer(s): 9