Fred, Cole and Ivan had a total of 43 stickers. The ratio of Cole's stickers to Ivan's stickers was 4 : 5 at first. Fred and Cole each gave away ¹₂ of their stickers. Given that the three boys had 29 stickers left, how many stickers did Fred have in the end?

	Fred	Cole	Ivan	Total
Comparing Cole and Ivan at first		4 u	5 u
Before	2 p	2x2 = 4 u	5 u	43
Change	- 1 p	-1x2 = - 2 u		- 14
After	1 p	1x2 = 2 u	5 u	29

Total number of stickers that Fred and Cole gave away
= 43 - 29
= 14

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

1 p + 2 u = 43 - 29
1 p + 2 u = 14
1 p = 14 - 2 u --- (1)

1 p + 2 u + 5 u = 29
1 p + 7 u = 29
1 p = 29 - 7 u --- (2)

(1) = (2)
14 - 2 u = 29 - 7 u
7 u - 2 u = 29 - 14
7 u - 2 u = 15
5 u = 15

1 u = 15 ÷ 5 = 3

Substitute 1 u = 3 into (1).
1 p = 14 - 2 u
1 p = 14 - 2 x 3
1 p = 14 - 6

1 p = 8

Number of stickers that Fred had in the end
= 1 p
= 8

Answer(s): 8