David, Xavier and Vincent had a total of 48 erasers. The ratio of Xavier's erasers to Vincent's erasers was 4 : 5 at first. David and Xavier each gave away ¹₂ of their erasers. Given that the three boys had 34 erasers left, how many erasers did David have in the end?

	David	Xavier	Vincent	Total
Comparing Xavier and Vincent at first		4 u	5 u
Before	2 p	2x2 = 4 u	5 u	48
Change	- 1 p	-1x2 = - 2 u		- 14
After	1 p	1x2 = 2 u	5 u	34

Total number of erasers that David and Xavier gave away
= 48 - 34
= 14

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

1 p + 2 u = 48 - 34
1 p + 2 u = 14
1 p = 14 - 2 u --- (1)

1 p + 2 u + 5 u = 34
1 p + 7 u = 34
1 p = 34 - 7 u --- (2)

(1) = (2)
14 - 2 u = 34 - 7 u
7 u - 2 u = 34 - 14
7 u - 2 u = 20
5 u = 20

1 u = 20 ÷ 5 = 4

Substitute 1 u = 4 into (1).
1 p = 14 - 2 u
1 p = 14 - 2 x 4
1 p = 14 - 8

1 p = 6

Number of erasers that David had in the end
= 1 p
= 6

Answer(s): 6