Perry, Seth and Will had a total of 112 pens. The ratio of Seth's pens to Will's pens was 4 : 9 at first. Perry and Seth each gave away ¹₂ of their pens. Given that the three boys had 92 pens left, how many pens did Perry have at first?

	Perry	Seth	Will	Total
Comparing Seth and Will at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	112
Change	- 1 p	-1x2 = - 2 u		- 20
After	1 p	1x2 = 2 u	9 u	92

Total number of pens that Perry and Seth gave away
= 112 - 92
= 20

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

1 p + 2 u = 112 - 92
1 p + 2 u = 20
1 p = 20 - 2 u --- (1)

1 p + 2 u + 9 u = 92
1 p + 11 u = 92
1 p = 92 - 11 u --- (2)

(1) = (2)
20 - 2 u = 92 - 11 u
11 u - 2 u = 92 - 20
11 u - 2 u = 72
9 u = 72

1 u = 72 ÷ 9 = 8

Substitute 1 u = 8 into (1).
1 p = 20 - 2 u
1 p = 20 - 2 x 8
1 p = 20 - 16

1 p = 4

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

Answer(s): 8