Owen, Fabian and Valen had a total of 54 beads. The ratio of Fabian's beads to Valen's beads was 8 : 9 at first. Owen and Fabian each gave away ¹₂ of their beads. Given that the three boys had 36 beads left, how many beads did Owen have in the end?

	Owen	Fabian	Valen	Total
Comparing Fabian and Valen at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	54
Change	- 1 p	-1x4 = - 4 u		- 18
After	1 p	1x4 = 4 u	9 u	36

Total number of beads that Owen and Fabian gave away
= 54 - 36
= 18

The number of beads that Fabian had at first is repeated.  Make the number of beads that Fabian had at first the same.  LCM of 8 and 2 is 8.

1 p + 4 u = 54 - 36
1 p + 4 u = 18
1 p = 18 - 4 u --- (1)

1 p + 4 u + 9 u = 36
1 p + 13 u = 36
1 p = 36 - 13 u --- (2)

(1) = (2)
18 - 4 u = 36 - 13 u
13 u - 4 u = 36 - 18
13 u - 4 u = 18
9 u = 18

1 u = 18 ÷ 9 = 2

Substitute 1 u = 2 into (1).
1 p = 18 - 4 u
1 p = 18 - 4 x 2
1 p = 18 - 8

1 p = 10

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

Answer(s): 10