Archie, Xavier and Billy had a total of 196 pencils. The ratio of Xavier's pencils to Billy's pencils was 8 : 7 at first. Archie and Xavier each gave away ¹₂ of their pencils. Given that the three boys had 140 pencils left, how many pencils did Archie have at first?

	Archie	Xavier	Billy	Total
Comparing Xavier and Billy at first		8 u	7 u
Before	2 p	2x4 = 8 u	7 u	196
Change	- 1 p	-1x4 = - 4 u		- 56
After	1 p	1x4 = 4 u	7 u	140

Total number of pencils that Archie and Xavier gave away
= 196 - 140
= 56

The number of pencils that Xavier had at first is repeated.  Make the number of pencils that Xavier had at first the same.  LCM of 8 and 2 is 8.

1 p + 4 u = 196 - 140
1 p + 4 u = 56
1 p = 56 - 4 u --- (1)

1 p + 4 u + 7 u = 140
1 p + 11 u = 140
1 p = 140 - 11 u --- (2)

(1) = (2)
56 - 4 u = 140 - 11 u
11 u - 4 u = 140 - 56
11 u - 4 u = 84
7 u = 84

1 u = 84 ÷ 7 = 12

Substitute 1 u = 12 into (1).
1 p = 56 - 4 u
1 p = 56 - 4 x 12
1 p = 56 - 48

1 p = 8

Number of pencils that Archie had at first
= 2 p
= 2 x 8
= 16

Answer(s): 16