Fabian, Jeremy and Ryan had a total of 69 pens. The ratio of Jeremy's pens to Ryan's pens was 8 : 3 at first. Fabian and Jeremy each gave away ¹₂ of their pens. Given that the three boys had 42 pens left, how many pens did Fabian have in the end?

	Fabian	Jeremy	Ryan	Total
Comparing Jeremy and Ryan at first		8 u	3 u
Before	2 p	2x4 = 8 u	3 u	69
Change	- 1 p	-1x4 = - 4 u		- 27
After	1 p	1x4 = 4 u	3 u	42

Total number of pens that Fabian and Jeremy gave away
= 69 - 42
= 27

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

1 p + 4 u = 69 - 42
1 p + 4 u = 27
1 p = 27 - 4 u --- (1)

1 p + 4 u + 3 u = 42
1 p + 7 u = 42
1 p = 42 - 7 u --- (2)

(1) = (2)
27 - 4 u = 42 - 7 u
7 u - 4 u = 42 - 27
7 u - 4 u = 15
3 u = 15

1 u = 15 ÷ 3 = 5

Substitute 1 u = 5 into (1).
1 p = 27 - 4 u
1 p = 27 - 4 x 5
1 p = 27 - 20

1 p = 7

Number of pens that Fabian had in the end
= 1 p
= 7

Answer(s): 7