Ryan, Justin and Julian had a total of 40 beads. The ratio of Justin's beads to Julian's beads was 4 : 3 at first. Ryan and Justin each gave away ¹₂ of their beads. Given that the three boys had 26 beads left, how many beads did Ryan have in the end?

	Ryan	Justin	Julian	Total
Comparing Justin and Julian at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	40
Change	- 1 p	-1x2 = - 2 u		- 14
After	1 p	1x2 = 2 u	3 u	26

Total number of beads that Ryan and Justin gave away
= 40 - 26
= 14

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

1 p + 2 u = 40 - 26
1 p + 2 u = 14
1 p = 14 - 2 u --- (1)

1 p + 2 u + 3 u = 26
1 p + 5 u = 26
1 p = 26 - 5 u --- (2)

(1) = (2)
14 - 2 u = 26 - 5 u
5 u - 2 u = 26 - 14
5 u - 2 u = 12
3 u = 12

1 u = 12 ÷ 3 = 4

Substitute 1 u = 4 into (1).
1 p = 14 - 2 u
1 p = 14 - 2 x 4
1 p = 14 - 8

1 p = 6

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

Answer(s): 6