Brandon, Zane and Will had a total of 144 beads. The ratio of Zane's beads to Will's beads was 6 : 5 at first. Brandon and Zane each gave away ¹₂ of their beads. Given that the three boys had 102 beads left, how many beads did Brandon have in the end?

	Brandon	Zane	Will	Total
Comparing Zane and Will at first		6 u	5 u
Before	2 p	2x3 = 6 u	5 u	144
Change	- 1 p	-1x3 = - 3 u		- 42
After	1 p	1x3 = 3 u	5 u	102

Total number of beads that Brandon and Zane gave away
= 144 - 102
= 42

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 6 and 2 is 6.

1 p + 3 u = 144 - 102
1 p + 3 u = 42
1 p = 42 - 3 u --- (1)

1 p + 3 u + 5 u = 102
1 p + 8 u = 102
1 p = 102 - 8 u --- (2)

(1) = (2)
42 - 3 u = 102 - 8 u
8 u - 3 u = 102 - 42
8 u - 3 u = 60
5 u = 60

1 u = 60 ÷ 5 = 12

Substitute 1 u = 12 into (1).
1 p = 42 - 3 u
1 p = 42 - 3 x 12
1 p = 42 - 36

1 p = 6

Number of beads that Brandon had in the end
= 1 p
= 6

Answer(s): 6