Dylan, Rael and Daniel had a total of 56 cards. The ratio of Rael's cards to Daniel's cards was 4 : 9 at first. Dylan and Rael each gave away ¹₂ of their cards. Given that the three boys had 46 cards left, how many cards did Dylan have in the end?

	Dylan	Rael	Daniel	Total
Comparing Rael and Daniel at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	56
Change	- 1 p	-1x2 = - 2 u		- 10
After	1 p	1x2 = 2 u	9 u	46

Total number of cards that Dylan and Rael gave away
= 56 - 46
= 10

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

1 p + 2 u = 56 - 46
1 p + 2 u = 10
1 p = 10 - 2 u --- (1)

1 p + 2 u + 9 u = 46
1 p + 11 u = 46
1 p = 46 - 11 u --- (2)

(1) = (2)
10 - 2 u = 46 - 11 u
11 u - 2 u = 46 - 10
11 u - 2 u = 36
9 u = 36

1 u = 36 ÷ 9 = 4

Substitute 1 u = 4 into (1).
1 p = 10 - 2 u
1 p = 10 - 2 x 4
1 p = 10 - 8

1 p = 2

Number of cards that Dylan had in the end
= 1 p
= 2

Answer(s): 2