Dylan, Riordan and Ivan had a total of 39 pencils. The ratio of Riordan's pencils to Ivan's pencils was 4 : 3 at first. Dylan and Riordan each gave away ¹₂ of their pencils. Given that the three boys had 27 pencils left, how many pencils did Dylan have in the end?

	Dylan	Riordan	Ivan	Total
Comparing Riordan and Ivan at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	39
Change	- 1 p	-1x2 = - 2 u		- 12
After	1 p	1x2 = 2 u	3 u	27

Total number of pencils that Dylan and Riordan gave away
= 39 - 27
= 12

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 = 39 - 27
1 p + 2 u = 12
1 p = 12 - 2 u --- (1)

1 p + 2 u + 3 u = 27
1 p + 5 u = 27
1 p = 27 - 5 u --- (2)

(1) = (2)
12 - 2 u = 27 - 5 u
5 u - 2 u = 27 - 12
5 u - 2 u = 15
3 u = 15

1 u = 15 ÷ 3 = 5

Substitute 1 u = 5 into (1).
1 p = 12 - 2 u
1 p = 12 - 2 x 5
1 p = 12 - 10

1 p = 2

Number of pencils that Dylan had in the end
= 1 p
= 2

Answer(s): 2