Paul, Tom and Ethan had a total of 36 cards. The ratio of Tom's cards to Ethan's cards was 8 : 7 at first. Paul and Tom each gave away ¹₂ of their cards. Given that the three boys had 25 cards left, how many cards did Paul have at first?

	Paul	Tom	Ethan	Total
Comparing Tom and Ethan at first		8 u	7 u
Before	2 p	2x4 = 8 u	7 u	36
Change	- 1 p	-1x4 = - 4 u		- 11
After	1 p	1x4 = 4 u	7 u	25

Total number of cards that Paul and Tom gave away
= 36 - 25
= 11

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 8 and 2 is 8.

1 p + 4 u = 36 - 25
1 p + 4 u = 11
1 p = 11 - 4 u --- (1)

1 p + 4 u + 7 u = 25
1 p + 11 u = 25
1 p = 25 - 11 u --- (2)

(1) = (2)
11 - 4 u = 25 - 11 u
11 u - 4 u = 25 - 11
11 u - 4 u = 14
7 u = 14

1 u = 14 ÷ 7 = 2

Substitute 1 u = 2 into (1).
1 p = 11 - 4 u
1 p = 11 - 4 x 2
1 p = 11 - 8

1 p = 3

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

Answer(s): 6