Fred, Daniel and Ryan had a total of 116 pens. The ratio of Daniel's pens to Ryan's pens was 8 : 5 at first. Fred and Daniel each gave away ¹₂ of their pens. Given that the three boys had 78 pens left, how many pens did Fred have in the end?

	Fred	Daniel	Ryan	Total
Comparing Daniel and Ryan at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	116
Change	- 1 p	-1x4 = - 4 u		- 38
After	1 p	1x4 = 4 u	5 u	78

Total number of pens that Fred and Daniel gave away
= 116 - 78
= 38

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

1 p + 4 u = 116 - 78
1 p + 4 u = 38
1 p = 38 - 4 u --- (1)

1 p + 4 u + 5 u = 78
1 p + 9 u = 78
1 p = 78 - 9 u --- (2)

(1) = (2)
38 - 4 u = 78 - 9 u
9 u - 4 u = 78 - 38
9 u - 4 u = 40
5 u = 40

1 u = 40 ÷ 5 = 8

Substitute 1 u = 8 into (1).
1 p = 38 - 4 u
1 p = 38 - 4 x 8
1 p = 38 - 32

1 p = 6

Number of pens that Fred had in the end
= 1 p
= 6

Answer(s): 6