Ethan, Luis and Glen had a total of 70 beads. The ratio of Luis's beads to Glen's beads was 6 : 5 at first. Ethan and Luis each gave away ¹₂ of their beads. Given that the three boys had 50 beads left, how many beads did Ethan have in the end?

	Ethan	Luis	Glen	Total
Comparing Luis and Glen at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	70
Change	- 1 p	-1x3 = - 3 u		- 20
After	1 p	1x3 = 3 u	5 u	50

Total number of beads that Ethan and Luis gave away
= 70 - 50
= 20

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

1 p + 3 u = 70 - 50
1 p + 3 u = 20
1 p = 20 - 3 u --- (1)

1 p + 3 u + 5 u = 50
1 p + 8 u = 50
1 p = 50 - 8 u --- (2)

(1) = (2)
20 - 3 u = 50 - 8 u
8 u - 3 u = 50 - 20
8 u - 3 u = 30
5 u = 30

1 u = 30 ÷ 5 = 6

Substitute 1 u = 6 into (1).
1 p = 20 - 3 u
1 p = 20 - 3 x 6
1 p = 20 - 18

1 p = 2

Number of beads that Ethan had in the end
= 1 p
= 2

Answer(s): 2