Nick, Jenson and Rael had a total of 57 beads. The ratio of Jenson's beads to Rael's beads was 4 : 3 at first. Nick and Jenson each gave away ¹₂ of their beads. Given that the three boys had 36 beads left, how many beads did Nick have at first?

	Nick	Jenson	Rael	Total
Comparing Jenson and Rael at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	57
Change	- 1 p	-1x2 = - 2 u		- 21
After	1 p	1x2 = 2 u	3 u	36

Total number of beads that Nick and Jenson gave away
= 57 - 36
= 21

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

1 p + 2 u = 57 - 36
1 p + 2 u = 21
1 p = 21 - 2 u --- (1)

1 p + 2 u + 3 u = 36
1 p + 5 u = 36
1 p = 36 - 5 u --- (2)

(1) = (2)
21 - 2 u = 36 - 5 u
5 u - 2 u = 36 - 21
5 u - 2 u = 15
3 u = 15

1 u = 15 ÷ 3 = 5

Substitute 1 u = 5 into (1).
1 p = 21 - 2 u
1 p = 21 - 2 x 5
1 p = 21 - 10

1 p = 11

Number of beads that Nick had at first
= 2 p
= 2 x 11
= 22

Answer(s): 22