Jenson, Billy and Perry had a total of 131 pencils. The ratio of Billy's pencils to Perry's pencils was 4 : 9 at first. Jenson and Billy each gave away ¹₂ of their pencils. Given that the three boys had 106 pencils left, how many pencils did Jenson have in the end?

	Jenson	Billy	Perry	Total
Comparing Billy and Perry at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	131
Change	- 1 p	-1x2 = - 2 u		- 25
After	1 p	1x2 = 2 u	9 u	106

Total number of pencils that Jenson and Billy gave away
= 131 - 106
= 25

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

1 p + 2 u = 131 - 106
1 p + 2 u = 25
1 p = 25 - 2 u --- (1)

1 p + 2 u + 9 u = 106
1 p + 11 u = 106
1 p = 106 - 11 u --- (2)

(1) = (2)
25 - 2 u = 106 - 11 u
11 u - 2 u = 106 - 25
11 u - 2 u = 81
9 u = 81

1 u = 81 ÷ 9 = 9

Substitute 1 u = 9 into (1).
1 p = 25 - 2 u
1 p = 25 - 2 x 9
1 p = 25 - 18

1 p = 7

Number of pencils that Jenson had in the end
= 1 p
= 7

Answer(s): 7