Fred, Dylan and Peter had a total of 36 cards. The ratio of Dylan's cards to Peter's cards was 2 : 5 at first. Fred and Dylan each gave away ¹₂ of their cards. Given that the three boys had 23 cards left, how many cards did Fred have at first?

	Fred	Dylan	Peter	Total
Comparing Dylan and Peter at first		2 u	5 u
Before	2 p	2x1 = 2 u	5 u	36
Change	- 1 p	-1x1 = - 1 u		- 13
After	1 p	1x1 = 1 u	5 u	23

Total number of cards that Fred and Dylan gave away
= 36 - 23
= 13

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

1 p + 1 u = 36 - 23
1 p + 1 u = 13
1 p = 13 - 1 u --- (1)

1 p + 1 u + 5 u = 23
1 p + 6 u = 23
1 p = 23 - 6 u --- (2)

(1) = (2)
13 - 1 u = 23 - 6 u
6 u - 1 u = 23 - 13
6 u - 1 u = 10
5 u = 10

1 u = 10 ÷ 5 = 2

Substitute 1 u = 2 into (1).
1 p = 13 - 1 u
1 p = 13 - 1 x 2
1 p = 13 - 2

1 p = 11

Number of cards that Fred had at first
= 2 p
= 2 x 11
= 22

Answer(s): 22