Valen, Xavier and Jack had a total of 30 pens. The ratio of Xavier's pens to Jack's pens was 2 : 9 at first. Valen and Xavier each gave away ¹₂ of their pens. Given that the three boys had 24 pens left, how many pens did Valen have at first?

	Valen	Xavier	Jack	Total
Comparing Xavier and Jack at first		2 u	9 u
Before	2 p	2x1 = 2 u	9 u	30
Change	- 1 p	-1x1 = - 1 u		- 6
After	1 p	1x1 = 1 u	9 u	24

Total number of pens that Valen and Xavier gave away
= 30 - 24
= 6

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

1 p + 1 u = 30 - 24
1 p + 1 u = 6
1 p = 6 - 1 u --- (1)

1 p + 1 u + 9 u = 24
1 p + 10 u = 24
1 p = 24 - 10 u --- (2)

(1) = (2)
6 - 1 u = 24 - 10 u
10 u - 1 u = 24 - 6
10 u - 1 u = 18
9 u = 18

1 u = 18 ÷ 9 = 2

Substitute 1 u = 2 into (1).
1 p = 6 - 1 u
1 p = 6 - 1 x 2
1 p = 6 - 2

1 p = 4

Number of pens that Valen had at first
= 2 p
= 2 x 4
= 8

Answer(s): 8