Bobby, Ethan and Zane had a total of 67 erasers. The ratio of Ethan's erasers to Zane's erasers was 2 : 5 at first. Bobby and Ethan each gave away ¹₂ of their erasers. Given that the three boys had 56 erasers left, how many erasers did Bobby have at first?

	Bobby	Ethan	Zane	Total
Comparing Ethan and Zane at first		2 u	5 u
Before	2 p	2x1 = 2 u	5 u	67
Change	- 1 p	-1x1 = - 1 u		- 11
After	1 p	1x1 = 1 u	5 u	56

Total number of erasers that Bobby and Ethan gave away
= 67 - 56
= 11

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

1 p + 1 u = 67 - 56
1 p + 1 u = 11
1 p = 11 - 1 u --- (1)

1 p + 1 u + 5 u = 56
1 p + 6 u = 56
1 p = 56 - 6 u --- (2)

(1) = (2)
11 - 1 u = 56 - 6 u
6 u - 1 u = 56 - 11
6 u - 1 u = 45
5 u = 45

1 u = 45 ÷ 5 = 9

Substitute 1 u = 9 into (1).
1 p = 11 - 1 u
1 p = 11 - 1 x 9
1 p = 11 - 9

1 p = 2

Number of erasers that Bobby had at first
= 2 p
= 2 x 2
= 4

Answer(s): 4