Harry, Ken and Oliver had a total of 57 cards. The ratio of Ken's cards to Oliver's cards was 4 : 3 at first. Harry and Ken each gave away ¹₂ of their cards. Given that the three boys had 39 cards left, how many cards did Harry have at first?

	Harry	Ken	Oliver	Total
Comparing Ken and Oliver at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	57
Change	- 1 p	-1x2 = - 2 u		- 18
After	1 p	1x2 = 2 u	3 u	39

Total number of cards that Harry and Ken gave away
= 57 - 39
= 18

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

1 p + 2 u = 57 - 39
1 p + 2 u = 18
1 p = 18 - 2 u --- (1)

1 p + 2 u + 3 u = 39
1 p + 5 u = 39
1 p = 39 - 5 u --- (2)

(1) = (2)
18 - 2 u = 39 - 5 u
5 u - 2 u = 39 - 18
5 u - 2 u = 21
3 u = 21

1 u = 21 ÷ 3 = 7

Substitute 1 u = 7 into (1).
1 p = 18 - 2 u
1 p = 18 - 2 x 7
1 p = 18 - 14

1 p = 4

Number of cards that Harry had at first
= 2 p
= 2 x 4
= 8

Answer(s): 8