Cole, Vincent and Henry had a total of 127 cards. The ratio of Vincent's cards to Henry's cards was 8 : 3 at first. Cole and Vincent each gave away ¹₂ of their cards. Given that the three boys had 80 cards left, how many cards did Cole have in the end?

	Cole	Vincent	Henry	Total
Comparing Vincent and Henry at first		8 u	3 u
Before	2 p	2x4 = 8 u	3 u	127
Change	- 1 p	-1x4 = - 4 u		- 47
After	1 p	1x4 = 4 u	3 u	80

Total number of cards that Cole and Vincent gave away
= 127 - 80
= 47

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 = 127 - 80
1 p + 4 u = 47
1 p = 47 - 4 u --- (1)

1 p + 4 u + 3 u = 80
1 p + 7 u = 80
1 p = 80 - 7 u --- (2)

(1) = (2)
47 - 4 u = 80 - 7 u
7 u - 4 u = 80 - 47
7 u - 4 u = 33
3 u = 33

1 u = 33 ÷ 3 = 11

Substitute 1 u = 11 into (1).
1 p = 47 - 4 u
1 p = 47 - 4 x 11
1 p = 47 - 44

1 p = 3

Number of cards that Cole had in the end
= 1 p
= 3

Answer(s): 3