Ethan, Caden and Ryan had a total of 68 pens. The ratio of Caden's pens to Ryan's pens was 2 : 7 at first. Ethan and Caden each gave away ¹₂ of their pens. Given that the three boys had 55 pens left, how many pens did Ethan have at first?

	Ethan	Caden	Ryan	Total
Comparing Caden and Ryan at first		2 u	7 u
Before	2 p	2x1 = 2 u	7 u	68
Change	- 1 p	-1x1 = - 1 u		- 13
After	1 p	1x1 = 1 u	7 u	55

Total number of pens that Ethan and Caden gave away
= 68 - 55
= 13

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

1 p + 1 u = 68 - 55
1 p + 1 u = 13
1 p = 13 - 1 u --- (1)

1 p + 1 u + 7 u = 55
1 p + 8 u = 55
1 p = 55 - 8 u --- (2)

(1) = (2)
13 - 1 u = 55 - 8 u
8 u - 1 u = 55 - 13
8 u - 1 u = 42
7 u = 42

1 u = 42 ÷ 7 = 6

Substitute 1 u = 6 into (1).
1 p = 13 - 1 u
1 p = 13 - 1 x 6
1 p = 13 - 6

1 p = 7

Number of pens that Ethan had at first
= 2 p
= 2 x 7
= 14

Answer(s): 14