Charlie, Elijah and Ryan had a total of 42 beads. The ratio of Elijah's beads to Ryan's beads was 6 : 5 at first. Charlie and Elijah each gave away ¹₂ of their beads. Given that the three boys had 26 beads left, how many beads did Charlie have in the end?

	Charlie	Elijah	Ryan	Total
Comparing Elijah and Ryan at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	42
Change	- 1 p	-1x3 = - 3 u		- 16
After	1 p	1x3 = 3 u	5 u	26

Total number of beads that Charlie and Elijah gave away
= 42 - 26
= 16

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

1 p + 3 u = 42 - 26
1 p + 3 u = 16
1 p = 16 - 3 u --- (1)

1 p + 3 u + 5 u = 26
1 p + 8 u = 26
1 p = 26 - 8 u --- (2)

(1) = (2)
16 - 3 u = 26 - 8 u
8 u - 3 u = 26 - 16
8 u - 3 u = 10
5 u = 10

1 u = 10 ÷ 5 = 2

Substitute 1 u = 2 into (1).
1 p = 16 - 3 u
1 p = 16 - 3 x 2
1 p = 16 - 6

1 p = 10

Number of beads that Charlie had in the end
= 1 p
= 10

Answer(s): 10