Jeremy, Zeph and Mark had a total of 146 pencils. The ratio of Zeph's pencils to Mark's pencils was 4 : 9 at first. Jeremy and Zeph each gave away ¹₂ of their pencils. Given that the three boys had 118 pencils left, how many pencils did Jeremy have at first?

	Jeremy	Zeph	Mark	Total
Comparing Zeph and Mark 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 pencils that Jeremy and Zeph gave away
= 146 - 118
= 28

The number of pencils that Zeph had at first is repeated.  Make the number of pencils that Zeph 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 pencils that Jeremy had at first
= 2 p
= 2 x 8
= 16

Answer(s): 16