Albert, Vincent and Cody had a total of 205 cards. The ratio of Vincent's cards to Cody's cards was 8 : 9 at first. Albert and Vincent each gave away ¹₂ of their cards. Given that the three boys had 152 cards left, how many cards did Albert have in the end?

	Albert	Vincent	Cody	Total
Comparing Vincent and Cody at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	205
Change	- 1 p	-1x4 = - 4 u		- 53
After	1 p	1x4 = 4 u	9 u	152

Total number of cards that Albert and Vincent gave away
= 205 - 152
= 53

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 8 and 2 is 8.

1 p + 4 u = 205 - 152
1 p + 4 u = 53
1 p = 53 - 4 u --- (1)

1 p + 4 u + 9 u = 152
1 p + 13 u = 152
1 p = 152 - 13 u --- (2)

(1) = (2)
53 - 4 u = 152 - 13 u
13 u - 4 u = 152 - 53
13 u - 4 u = 99
9 u = 99

1 u = 99 ÷ 9 = 11

Substitute 1 u = 11 into (1).
1 p = 53 - 4 u
1 p = 53 - 4 x 11
1 p = 53 - 44

1 p = 9

Number of cards that Albert had in the end
= 1 p
= 9

Answer(s): 9