Michael, Lee and Carl had a total of 37 cards. The ratio of Lee's cards to Carl's cards was 2 : 7 at first. Michael and Lee each gave away ¹₂ of their cards. Given that the three boys had 29 cards left, how many cards did Michael have at first?

	Michael	Lee	Carl	Total
Comparing Lee and Carl at first		2 u	7 u
Before	2 p	2x1 = 2 u	7 u	37
Change	- 1 p	-1x1 = - 1 u		- 8
After	1 p	1x1 = 1 u	7 u	29

Total number of cards that Michael and Lee gave away
= 37 - 29
= 8

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

1 p + 1 u = 37 - 29
1 p + 1 u = 8
1 p = 8 - 1 u --- (1)

1 p + 1 u + 7 u = 29
1 p + 8 u = 29
1 p = 29 - 8 u --- (2)

(1) = (2)
8 - 1 u = 29 - 8 u
8 u - 1 u = 29 - 8
8 u - 1 u = 21
7 u = 21

1 u = 21 ÷ 7 = 3

Substitute 1 u = 3 into (1).
1 p = 8 - 1 u
1 p = 8 - 1 x 3
1 p = 8 - 3

1 p = 5

Number of cards that Michael had at first
= 2 p
= 2 x 5
= 10

Answer(s): 10