Sam, Michael and Luke had a total of 178 pens. The ratio of Michael's pens to Luke's pens was 4 : 9 at first. Sam and Michael each gave away ¹₂ of their pens. Given that the three boys had 143 pens left, how many pens did Sam have in the end?

	Sam	Michael	Luke	Total
Comparing Michael and Luke at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	178
Change	- 1 p	-1x2 = - 2 u		- 35
After	1 p	1x2 = 2 u	9 u	143

Total number of pens that Sam and Michael gave away
= 178 - 143
= 35

The number of pens that Michael had at first is repeated.  Make the number of pens that Michael had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 178 - 143
1 p + 2 u = 35
1 p = 35 - 2 u --- (1)

1 p + 2 u + 9 u = 143
1 p + 11 u = 143
1 p = 143 - 11 u --- (2)

(1) = (2)
35 - 2 u = 143 - 11 u
11 u - 2 u = 143 - 35
11 u - 2 u = 108
9 u = 108

1 u = 108 ÷ 9 = 12

Substitute 1 u = 12 into (1).
1 p = 35 - 2 u
1 p = 35 - 2 x 12
1 p = 35 - 24

1 p = 11

Number of pens that Sam had in the end
= 1 p
= 11

Answer(s): 11