Liam, Rael and Japheth had a total of 146 pens. The ratio of Rael's pens to Japheth's pens was 4 : 9 at first. Liam and Rael each gave away ¹₂ of their pens. Given that the three boys had 118 pens left, how many pens did Liam have in the end?

	Liam	Rael	Japheth	Total
Comparing Rael and Japheth at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	146
Change	- 1 p	-1x2 = - 2 u		- 28
After	1 p	1x2 = 2 u	9 u	118

Total number of pens that Liam and Rael gave away
= 146 - 118
= 28

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

1 p + 2 u = 146 - 118
1 p + 2 u = 28
1 p = 28 - 2 u --- (1)

1 p + 2 u + 9 u = 118
1 p + 11 u = 118
1 p = 118 - 11 u --- (2)

(1) = (2)
28 - 2 u = 118 - 11 u
11 u - 2 u = 118 - 28
11 u - 2 u = 90
9 u = 90

1 u = 90 ÷ 9 = 10

Substitute 1 u = 10 into (1).
1 p = 28 - 2 u
1 p = 28 - 2 x 10
1 p = 28 - 20

1 p = 8

Number of pens that Liam had in the end
= 1 p
= 8

Answer(s): 8