Japheth, Sean and Vincent had a total of 106 beads. The ratio of Sean's beads to Vincent's beads was 8 : 3 at first. Japheth and Sean each gave away ¹₂ of their beads. Given that the three boys had 65 beads left, how many beads did Japheth have at first?

	Japheth	Sean	Vincent	Total
Comparing Sean and Vincent at first		8 u	3 u
Before	2 p	2x4 = 8 u	3 u	106
Change	- 1 p	-1x4 = - 4 u		- 41
After	1 p	1x4 = 4 u	3 u	65

Total number of beads that Japheth and Sean gave away
= 106 - 65
= 41

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

1 p + 4 u = 106 - 65
1 p + 4 u = 41
1 p = 41 - 4 u --- (1)

1 p + 4 u + 3 u = 65
1 p + 7 u = 65
1 p = 65 - 7 u --- (2)

(1) = (2)
41 - 4 u = 65 - 7 u
7 u - 4 u = 65 - 41
7 u - 4 u = 24
3 u = 24

1 u = 24 ÷ 3 = 8

Substitute 1 u = 8 into (1).
1 p = 41 - 4 u
1 p = 41 - 4 x 8
1 p = 41 - 32

1 p = 9

Number of beads that Japheth had at first
= 2 p
= 2 x 9
= 18

Answer(s): 18