David, Eric and Cole had a total of 173 pens. The ratio of Eric's pens to Cole's pens was 8 : 9 at first. David and Eric each gave away ¹₂ of their pens. Given that the three boys had 127 pens left, how many pens did David have in the end?

	David	Eric	Cole	Total
Comparing Eric and Cole at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	173
Change	- 1 p	-1x4 = - 4 u		- 46
After	1 p	1x4 = 4 u	9 u	127

Total number of pens that David and Eric gave away
= 173 - 127
= 46

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

1 p + 4 u = 173 - 127
1 p + 4 u = 46
1 p = 46 - 4 u --- (1)

1 p + 4 u + 9 u = 127
1 p + 13 u = 127
1 p = 127 - 13 u --- (2)

(1) = (2)
46 - 4 u = 127 - 13 u
13 u - 4 u = 127 - 46
13 u - 4 u = 81
9 u = 81

1 u = 81 ÷ 9 = 9

Substitute 1 u = 9 into (1).
1 p = 46 - 4 u
1 p = 46 - 4 x 9
1 p = 46 - 36

1 p = 10

Number of pens that David had in the end
= 1 p
= 10

Answer(s): 10