Archie, Ryan and Mark had a total of 28 pens. The ratio of Ryan's pens to Mark's pens was 4 : 5 at first. Archie and Ryan each gave away ¹₂ of their pens. Given that the three boys had 19 pens left, how many pens did Archie have at first?

	Archie	Ryan	Mark	Total
Comparing Ryan and Mark at first		4 u	5 u
Before	2 p	2x2 = 4 u	5 u	28
Change	- 1 p	-1x2 = - 2 u		- 9
After	1 p	1x2 = 2 u	5 u	19

Total number of pens that Archie and Ryan gave away
= 28 - 19
= 9

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

1 p + 2 u = 28 - 19
1 p + 2 u = 9
1 p = 9 - 2 u --- (1)

1 p + 2 u + 5 u = 19
1 p + 7 u = 19
1 p = 19 - 7 u --- (2)

(1) = (2)
9 - 2 u = 19 - 7 u
7 u - 2 u = 19 - 9
7 u - 2 u = 10
5 u = 10

1 u = 10 ÷ 5 = 2

Substitute 1 u = 2 into (1).
1 p = 9 - 2 u
1 p = 9 - 2 x 2
1 p = 9 - 4

1 p = 5

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

Answer(s): 10