Ethan, Tim and Gabriel had a total of 52 stickers. The ratio of Tim's stickers to Gabriel's stickers was 4 : 5 at first. Ethan and Tim each gave away ¹₂ of their stickers. Given that the three boys had 36 stickers left, how many stickers did Ethan have at first?

	Ethan	Tim	Gabriel	Total
Comparing Tim and Gabriel at first		4 u	5 u
Before	2 p	2x2 = 4 u	5 u	52
Change	- 1 p	-1x2 = - 2 u		- 16
After	1 p	1x2 = 2 u	5 u	36

Total number of stickers that Ethan and Tim gave away
= 52 - 36
= 16

The number of stickers that Tim had at first is repeated.  Make the number of stickers that Tim had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 52 - 36
1 p + 2 u = 16
1 p = 16 - 2 u --- (1)

1 p + 2 u + 5 u = 36
1 p + 7 u = 36
1 p = 36 - 7 u --- (2)

(1) = (2)
16 - 2 u = 36 - 7 u
7 u - 2 u = 36 - 16
7 u - 2 u = 20
5 u = 20

1 u = 20 ÷ 5 = 4

Substitute 1 u = 4 into (1).
1 p = 16 - 2 u
1 p = 16 - 2 x 4
1 p = 16 - 8

1 p = 8

Number of stickers that Ethan had at first
= 2 p
= 2 x 8
= 16

Answer(s): 16