Sean, Tommy and Elijah had a total of 44 pens. The ratio of Tommy's pens to Elijah's pens was 2 : 3 at first. Sean and Tommy each gave away ¹₂ of their pens. Given that the three boys had 34 pens left, how many pens did Sean have in the end?

	Sean	Tommy	Elijah	Total
Comparing Tommy and Elijah at first		2 u	3 u
Before	2 p	2x1 = 2 u	3 u	44
Change	- 1 p	-1x1 = - 1 u		- 10
After	1 p	1x1 = 1 u	3 u	34

Total number of pens that Sean and Tommy gave away
= 44 - 34
= 10

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

1 p + 1 u = 44 - 34
1 p + 1 u = 10
1 p = 10 - 1 u --- (1)

1 p + 1 u + 3 u = 34
1 p + 4 u = 34
1 p = 34 - 4 u --- (2)

(1) = (2)
10 - 1 u = 34 - 4 u
4 u - 1 u = 34 - 10
4 u - 1 u = 24
3 u = 24

1 u = 24 ÷ 3 = 8

Substitute 1 u = 8 into (1).
1 p = 10 - 1 u
1 p = 10 - 1 x 8
1 p = 10 - 8

1 p = 2

Number of pens that Sean had in the end
= 1 p
= 2

Answer(s): 2