Cody, Vincent and Charlie had a total of 159 cards. The ratio of Vincent's cards to Charlie's cards was 4 : 9 at first. Cody and Vincent each gave away ¹₂ of their cards. Given that the three boys had 129 cards left, how many cards did Cody have at first?

	Cody	Vincent	Charlie	Total
Comparing Vincent and Charlie at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	159
Change	- 1 p	-1x2 = - 2 u		- 30
After	1 p	1x2 = 2 u	9 u	129

Total number of cards that Cody and Vincent gave away
= 159 - 129
= 30

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

1 p + 2 u = 159 - 129
1 p + 2 u = 30
1 p = 30 - 2 u --- (1)

1 p + 2 u + 9 u = 129
1 p + 11 u = 129
1 p = 129 - 11 u --- (2)

(1) = (2)
30 - 2 u = 129 - 11 u
11 u - 2 u = 129 - 30
11 u - 2 u = 99
9 u = 99

1 u = 99 ÷ 9 = 11

Substitute 1 u = 11 into (1).
1 p = 30 - 2 u
1 p = 30 - 2 x 11
1 p = 30 - 22

1 p = 8

Number of cards that Cody had at first
= 2 p
= 2 x 8
= 16

Answer(s): 16