Bryan, Riordan and George had a total of 52 pens. The ratio of Riordan's pens to George's pens was 4 : 3 at first. Bryan and Riordan each gave away ¹₂ of their pens. Given that the three boys had 32 pens left, how many pens did Bryan have at first?

	Bryan	Riordan	George	Total
Comparing Riordan and George at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	52
Change	- 1 p	-1x2 = - 2 u		- 20
After	1 p	1x2 = 2 u	3 u	32

Total number of pens that Bryan and Riordan gave away
= 52 - 32
= 20

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

1 p + 2 u = 52 - 32
1 p + 2 u = 20
1 p = 20 - 2 u --- (1)

1 p + 2 u + 3 u = 32
1 p + 5 u = 32
1 p = 32 - 5 u --- (2)

(1) = (2)
20 - 2 u = 32 - 5 u
5 u - 2 u = 32 - 20
5 u - 2 u = 12
3 u = 12

1 u = 12 ÷ 3 = 4

Substitute 1 u = 4 into (1).
1 p = 20 - 2 u
1 p = 20 - 2 x 4
1 p = 20 - 8

1 p = 12

Number of pens that Bryan had at first
= 2 p
= 2 x 12
= 24

Answer(s): 24