Seth, Ian and Michael had a total of 64 pens. The ratio of Ian's pens to Michael's pens was 6 : 7 at first. Seth and Ian each gave away ¹₂ of their pens. Given that the three boys had 46 pens left, how many pens did Seth have in the end?

	Seth	Ian	Michael	Total
Comparing Ian and Michael at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	64
Change	- 1 p	-1x3 = - 3 u		- 18
After	1 p	1x3 = 3 u	7 u	46

Total number of pens that Seth and Ian gave away
= 64 - 46
= 18

The number of pens that Ian had at first is repeated.  Make the number of pens that Ian had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 64 - 46
1 p + 3 u = 18
1 p = 18 - 3 u --- (1)

1 p + 3 u + 7 u = 46
1 p + 10 u = 46
1 p = 46 - 10 u --- (2)

(1) = (2)
18 - 3 u = 46 - 10 u
10 u - 3 u = 46 - 18
10 u - 3 u = 28
7 u = 28

1 u = 28 ÷ 7 = 4

Substitute 1 u = 4 into (1).
1 p = 18 - 3 u
1 p = 18 - 3 x 4
1 p = 18 - 12

1 p = 6

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

Answer(s): 6