Henry, Albert and Tommy had a total of 61 cards. The ratio of Albert's cards to Tommy's cards was 4 : 9 at first. Henry and Albert each gave away ¹₂ of their cards. Given that the three boys had 44 cards left, how many cards did Henry have in the end?

	Henry	Albert	Tommy	Total
Comparing Albert and Tommy at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	61
Change	- 1 p	-1x2 = - 2 u		- 17
After	1 p	1x2 = 2 u	9 u	44

Total number of cards that Henry and Albert gave away
= 61 - 44
= 17

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

1 p + 2 u = 61 - 44
1 p + 2 u = 17
1 p = 17 - 2 u --- (1)

1 p + 2 u + 9 u = 44
1 p + 11 u = 44
1 p = 44 - 11 u --- (2)

(1) = (2)
17 - 2 u = 44 - 11 u
11 u - 2 u = 44 - 17
11 u - 2 u = 27
9 u = 27

1 u = 27 ÷ 9 = 3

Substitute 1 u = 3 into (1).
1 p = 17 - 2 u
1 p = 17 - 2 x 3
1 p = 17 - 6

1 p = 11

Number of cards that Henry had in the end
= 1 p
= 11

Answer(s): 11