Valen, Simon and Albert had a total of 140 pens. The ratio of Simon's pens to Albert's pens was 4 : 9 at first. Valen and Simon each gave away ¹₂ of their pens. Given that the three boys had 115 pens left, how many pens did Valen have at first?

	Valen	Simon	Albert	Total
Comparing Simon and Albert at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	140
Change	- 1 p	-1x2 = - 2 u		- 25
After	1 p	1x2 = 2 u	9 u	115

Total number of pens that Valen and Simon gave away
= 140 - 115
= 25

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

1 p + 2 u = 140 - 115
1 p + 2 u = 25
1 p = 25 - 2 u --- (1)

1 p + 2 u + 9 u = 115
1 p + 11 u = 115
1 p = 115 - 11 u --- (2)

(1) = (2)
25 - 2 u = 115 - 11 u
11 u - 2 u = 115 - 25
11 u - 2 u = 90
9 u = 90

1 u = 90 ÷ 9 = 10

Substitute 1 u = 10 into (1).
1 p = 25 - 2 u
1 p = 25 - 2 x 10
1 p = 25 - 20

1 p = 5

Number of pens that Valen had at first
= 2 p
= 2 x 5
= 10

Answer(s): 10