Rael, Zane and Glen had a total of 57 beads. The ratio of Zane's beads to Glen's beads was 4 : 7 at first. Rael and Zane each gave away ¹₂ of their beads. Given that the three boys had 39 beads left, how many beads did Rael have in the end?

	Rael	Zane	Glen	Total
Comparing Zane and Glen at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	57
Change	- 1 p	-1x2 = - 2 u		- 18
After	1 p	1x2 = 2 u	7 u	39

Total number of beads that Rael and Zane gave away
= 57 - 39
= 18

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

1 p + 2 u = 57 - 39
1 p + 2 u = 18
1 p = 18 - 2 u --- (1)

1 p + 2 u + 7 u = 39
1 p + 9 u = 39
1 p = 39 - 9 u --- (2)

(1) = (2)
18 - 2 u = 39 - 9 u
9 u - 2 u = 39 - 18
9 u - 2 u = 21
7 u = 21

1 u = 21 ÷ 7 = 3

Substitute 1 u = 3 into (1).
1 p = 18 - 2 u
1 p = 18 - 2 x 3
1 p = 18 - 6

1 p = 12

Number of beads that Rael had in the end
= 1 p
= 12

Answer(s): 12