Elijah, Jeremy and Neave had a total of 24 cards. The ratio of Jeremy's cards to Neave's cards was 2 : 3 at first. Elijah and Jeremy each gave away ¹₂ of their cards. Given that the three boys had 15 cards left, how many cards did Elijah have in the end?

	Elijah	Jeremy	Neave	Total
Comparing Jeremy and Neave at first		2 u	3 u
Before	2 p	2x1 = 2 u	3 u	24
Change	- 1 p	-1x1 = - 1 u		- 9
After	1 p	1x1 = 1 u	3 u	15

Total number of cards that Elijah and Jeremy gave away
= 24 - 15
= 9

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

1 p + 1 u = 24 - 15
1 p + 1 u = 9
1 p = 9 - 1 u --- (1)

1 p + 1 u + 3 u = 15
1 p + 4 u = 15
1 p = 15 - 4 u --- (2)

(1) = (2)
9 - 1 u = 15 - 4 u
4 u - 1 u = 15 - 9
4 u - 1 u = 6
3 u = 6

1 u = 6 ÷ 3 = 2

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

1 p = 7

Number of cards that Elijah had in the end
= 1 p
= 7

Answer(s): 7