Brandon, Owen and Carl had a total of 134 cards. The ratio of Owen's cards to Carl's cards was 2 : 9 at first. Brandon and Owen each gave away ¹₂ of their cards. Given that the three boys had 112 cards left, how many cards did Brandon have at first?

	Brandon	Owen	Carl	Total
Comparing Owen and Carl at first		2 u	9 u
Before	2 p	2x1 = 2 u	9 u	134
Change	- 1 p	-1x1 = - 1 u		- 22
After	1 p	1x1 = 1 u	9 u	112

Total number of cards that Brandon and Owen gave away
= 134 - 112
= 22

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

1 p + 1 u = 134 - 112
1 p + 1 u = 22
1 p = 22 - 1 u --- (1)

1 p + 1 u + 9 u = 112
1 p + 10 u = 112
1 p = 112 - 10 u --- (2)

(1) = (2)
22 - 1 u = 112 - 10 u
10 u - 1 u = 112 - 22
10 u - 1 u = 90
9 u = 90

1 u = 90 ÷ 9 = 10

Substitute 1 u = 10 into (1).
1 p = 22 - 1 u
1 p = 22 - 1 x 10
1 p = 22 - 10

1 p = 12

Number of cards that Brandon had at first
= 2 p
= 2 x 12
= 24

Answer(s): 24