Seth, Ryan and Reggie had a total of 144 cards. The ratio of Ryan's cards to Reggie's cards was 4 : 7 at first. Seth and Ryan each gave away ¹₂ of their cards. Given that the three boys had 114 cards left, how many cards did Seth have in the end?

	Seth	Ryan	Reggie	Total
Comparing Ryan and Reggie at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	144
Change	- 1 p	-1x2 = - 2 u		- 30
After	1 p	1x2 = 2 u	7 u	114

Total number of cards that Seth and Ryan gave away
= 144 - 114
= 30

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

1 p + 2 u = 144 - 114
1 p + 2 u = 30
1 p = 30 - 2 u --- (1)

1 p + 2 u + 7 u = 114
1 p + 9 u = 114
1 p = 114 - 9 u --- (2)

(1) = (2)
30 - 2 u = 114 - 9 u
9 u - 2 u = 114 - 30
9 u - 2 u = 84
7 u = 84

1 u = 84 ÷ 7 = 12

Substitute 1 u = 12 into (1).
1 p = 30 - 2 u
1 p = 30 - 2 x 12
1 p = 30 - 24

1 p = 6

Number of cards that Seth had in the end
= 1 p
= 6

Answer(s): 6