Daniel, Tom and Riordan had a total of 116 cards. The ratio of Tom's cards to Riordan's cards was 4 : 7 at first. Daniel and Tom each gave away ¹₂ of their cards. Given that the three boys had 93 cards left, how many cards did Daniel have at first?

	Daniel	Tom	Riordan	Total
Comparing Tom and Riordan at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	116
Change	- 1 p	-1x2 = - 2 u		- 23
After	1 p	1x2 = 2 u	7 u	93

Total number of cards that Daniel and Tom gave away
= 116 - 93
= 23

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

1 p + 2 u = 116 - 93
1 p + 2 u = 23
1 p = 23 - 2 u --- (1)

1 p + 2 u + 7 u = 93
1 p + 9 u = 93
1 p = 93 - 9 u --- (2)

(1) = (2)
23 - 2 u = 93 - 9 u
9 u - 2 u = 93 - 23
9 u - 2 u = 70
7 u = 70

1 u = 70 ÷ 7 = 10

Substitute 1 u = 10 into (1).
1 p = 23 - 2 u
1 p = 23 - 2 x 10
1 p = 23 - 20

1 p = 3

Number of cards that Daniel had at first
= 2 p
= 2 x 3
= 6

Answer(s): 6