Caden, Flynn and Luke had a total of 30 marbles. The ratio of Flynn's marbles to Luke's marbles was 2 : 3 at first. Caden and Flynn each gave away ¹₂ of their marbles. Given that the three boys had 18 marbles left, how many marbles did Caden have at first?

	Caden	Flynn	Luke	Total
Comparing Flynn and Luke at first		2 u	3 u
Before	2 p	2x1 = 2 u	3 u	30
Change	- 1 p	-1x1 = - 1 u		- 12
After	1 p	1x1 = 1 u	3 u	18

Total number of marbles that Caden and Flynn gave away
= 30 - 18
= 12

The number of marbles that Flynn had at first is repeated.  Make the number of marbles that Flynn had at first the same.  LCM of 2 and 2 is 2.

1 p + 1 u = 30 - 18
1 p + 1 u = 12
1 p = 12 - 1 u --- (1)

1 p + 1 u + 3 u = 18
1 p + 4 u = 18
1 p = 18 - 4 u --- (2)

(1) = (2)
12 - 1 u = 18 - 4 u
4 u - 1 u = 18 - 12
4 u - 1 u = 6
3 u = 6

1 u = 6 ÷ 3 = 2

Substitute 1 u = 2 into (1).
1 p = 12 - 1 u
1 p = 12 - 1 x 2
1 p = 12 - 2

1 p = 10

Number of marbles that Caden had at first
= 2 p
= 2 x 10
= 20

Answer(s): 20