Flynn, Jenson and Japheth had a total of 46 cards. The ratio of Jenson's cards to Japheth's cards was 6 : 7 at first. Flynn and Jenson each gave away ¹₂ of their cards. Given that the three boys had 30 cards left, how many cards did Flynn have at first?

	Flynn	Jenson	Japheth	Total
Comparing Jenson and Japheth at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	46
Change	- 1 p	-1x3 = - 3 u		- 16
After	1 p	1x3 = 3 u	7 u	30

Total number of cards that Flynn and Jenson gave away
= 46 - 30
= 16

The number of cards that Jenson had at first is repeated.  Make the number of cards that Jenson had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 46 - 30
1 p + 3 u = 16
1 p = 16 - 3 u --- (1)

1 p + 3 u + 7 u = 30
1 p + 10 u = 30
1 p = 30 - 10 u --- (2)

(1) = (2)
16 - 3 u = 30 - 10 u
10 u - 3 u = 30 - 16
10 u - 3 u = 14
7 u = 14

1 u = 14 ÷ 7 = 2

Substitute 1 u = 2 into (1).
1 p = 16 - 3 u
1 p = 16 - 3 x 2
1 p = 16 - 6

1 p = 10

Number of cards that Flynn had at first
= 2 p
= 2 x 10
= 20

Answer(s): 20