Peter, Cody and Japheth had a total of 166 pens. The ratio of Cody's pens to Japheth's pens was 4 : 9 at first. Peter and Cody each gave away ¹₂ of their pens. Given that the three boys had 137 pens left, how many pens did Peter have at first?

	Peter	Cody	Japheth	Total
Comparing Cody and Japheth at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	166
Change	- 1 p	-1x2 = - 2 u		- 29
After	1 p	1x2 = 2 u	9 u	137

Total number of pens that Peter and Cody gave away
= 166 - 137
= 29

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

1 p + 2 u = 166 - 137
1 p + 2 u = 29
1 p = 29 - 2 u --- (1)

1 p + 2 u + 9 u = 137
1 p + 11 u = 137
1 p = 137 - 11 u --- (2)

(1) = (2)
29 - 2 u = 137 - 11 u
11 u - 2 u = 137 - 29
11 u - 2 u = 108
9 u = 108

1 u = 108 ÷ 9 = 12

Substitute 1 u = 12 into (1).
1 p = 29 - 2 u
1 p = 29 - 2 x 12
1 p = 29 - 24

1 p = 5

Number of pens that Peter had at first
= 2 p
= 2 x 5
= 10

Answer(s): 10