Reggie, Albert and Ivan had a total of 51 erasers. The ratio of Albert's erasers to Ivan's erasers was 6 : 7 at first. Reggie and Albert each gave away ¹₂ of their erasers. Given that the three boys had 36 erasers left, how many erasers did Reggie have in the end?

	Reggie	Albert	Ivan	Total
Comparing Albert and Ivan at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	51
Change	- 1 p	-1x3 = - 3 u		- 15
After	1 p	1x3 = 3 u	7 u	36

Total number of erasers that Reggie and Albert gave away
= 51 - 36
= 15

The number of erasers that Albert had at first is repeated.  Make the number of erasers that Albert had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 51 - 36
1 p + 3 u = 15
1 p = 15 - 3 u --- (1)

1 p + 3 u + 7 u = 36
1 p + 10 u = 36
1 p = 36 - 10 u --- (2)

(1) = (2)
15 - 3 u = 36 - 10 u
10 u - 3 u = 36 - 15
10 u - 3 u = 21
7 u = 21

1 u = 21 ÷ 7 = 3

Substitute 1 u = 3 into (1).
1 p = 15 - 3 u
1 p = 15 - 3 x 3
1 p = 15 - 9

1 p = 6

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

Answer(s): 6