Julian, Wesley and Bryan had a total of 90 cards. The ratio of Wesley's cards to Bryan's cards was 4 : 5 at first. Julian and Wesley each gave away ¹₂ of their cards. Given that the three boys had 65 cards left, how many cards did Julian have at first?

	Julian	Wesley	Bryan	Total
Comparing Wesley and Bryan at first		4 u	5 u
Before	2 p	2x2 = 4 u	5 u	90
Change	- 1 p	-1x2 = - 2 u		- 25
After	1 p	1x2 = 2 u	5 u	65

Total number of cards that Julian and Wesley gave away
= 90 - 65
= 25

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

1 p + 2 u = 90 - 65
1 p + 2 u = 25
1 p = 25 - 2 u --- (1)

1 p + 2 u + 5 u = 65
1 p + 7 u = 65
1 p = 65 - 7 u --- (2)

(1) = (2)
25 - 2 u = 65 - 7 u
7 u - 2 u = 65 - 25
7 u - 2 u = 40
5 u = 40

1 u = 40 ÷ 5 = 8

Substitute 1 u = 8 into (1).
1 p = 25 - 2 u
1 p = 25 - 2 x 8
1 p = 25 - 16

1 p = 9

Number of cards that Julian had at first
= 2 p
= 2 x 9
= 18

Answer(s): 18