Simon, George and Valen had a total of 32 pens. The ratio of George's pens to Valen's pens was 8 : 5 at first. Simon and George each gave away ¹₂ of their pens. Given that the three boys had 21 pens left, how many pens did Simon have at first?

	Simon	George	Valen	Total
Comparing George and Valen at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	32
Change	- 1 p	-1x4 = - 4 u		- 11
After	1 p	1x4 = 4 u	5 u	21

Total number of pens that Simon and George gave away
= 32 - 21
= 11

The number of pens that George had at first is repeated.  Make the number of pens that George had at first the same.  LCM of 8 and 2 is 8.

1 p + 4 u = 32 - 21
1 p + 4 u = 11
1 p = 11 - 4 u --- (1)

1 p + 4 u + 5 u = 21
1 p + 9 u = 21
1 p = 21 - 9 u --- (2)

(1) = (2)
11 - 4 u = 21 - 9 u
9 u - 4 u = 21 - 11
9 u - 4 u = 10
5 u = 10

1 u = 10 ÷ 5 = 2

Substitute 1 u = 2 into (1).
1 p = 11 - 4 u
1 p = 11 - 4 x 2
1 p = 11 - 8

1 p = 3

Number of pens that Simon had at first
= 2 p
= 2 x 3
= 6

Answer(s): 6