Jack, Riordan and Billy had a total of 91 pencils. The ratio of Riordan's pencils to Billy's pencils was 4 : 3 at first. Jack and Riordan each gave away ¹₂ of their pencils. Given that the three boys had 62 pencils left, how many pencils did Jack have at first?

	Jack	Riordan	Billy	Total
Comparing Riordan and Billy at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	91
Change	- 1 p	-1x2 = - 2 u		- 29
After	1 p	1x2 = 2 u	3 u	62

Total number of pencils that Jack and Riordan gave away
= 91 - 62
= 29

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

1 p + 2 u = 91 - 62
1 p + 2 u = 29
1 p = 29 - 2 u --- (1)

1 p + 2 u + 3 u = 62
1 p + 5 u = 62
1 p = 62 - 5 u --- (2)

(1) = (2)
29 - 2 u = 62 - 5 u
5 u - 2 u = 62 - 29
5 u - 2 u = 33
3 u = 33

1 u = 33 ÷ 3 = 11

Substitute 1 u = 11 into (1).
1 p = 29 - 2 u
1 p = 29 - 2 x 11
1 p = 29 - 22

1 p = 7

Number of pencils that Jack had at first
= 2 p
= 2 x 7
= 14

Answer(s): 14