Sean, Fred and Asher had a total of 89 cards. The ratio of Fred's cards to Asher's cards was 4 : 5 at first. Sean and Fred each gave away ¹₂ of their cards. Given that the three boys had 67 cards left, how many cards did Sean have in the end?

	Sean	Fred	Asher	Total
Comparing Fred and Asher at first		4 u	5 u
Before	2 p	2x2 = 4 u	5 u	89
Change	- 1 p	-1x2 = - 2 u		- 22
After	1 p	1x2 = 2 u	5 u	67

Total number of cards that Sean and Fred gave away
= 89 - 67
= 22

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

1 p + 2 u = 89 - 67
1 p + 2 u = 22
1 p = 22 - 2 u --- (1)

1 p + 2 u + 5 u = 67
1 p + 7 u = 67
1 p = 67 - 7 u --- (2)

(1) = (2)
22 - 2 u = 67 - 7 u
7 u - 2 u = 67 - 22
7 u - 2 u = 45
5 u = 45

1 u = 45 ÷ 5 = 9

Substitute 1 u = 9 into (1).
1 p = 22 - 2 u
1 p = 22 - 2 x 9
1 p = 22 - 18

1 p = 4

Number of cards that Sean had in the end
= 1 p
= 4

Answer(s): 4