Justin, Lee and Fred had a total of 148 erasers. The ratio of Lee's erasers to Fred's erasers was 8 : 9 at first. Justin and Lee each gave away ¹₂ of their erasers. Given that the three boys had 110 erasers left, how many erasers did Justin have at first?

	Justin	Lee	Fred	Total
Comparing Lee and Fred at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	148
Change	- 1 p	-1x4 = - 4 u		- 38
After	1 p	1x4 = 4 u	9 u	110

Total number of erasers that Justin and Lee gave away
= 148 - 110
= 38

The number of erasers that Lee had at first is repeated.  Make the number of erasers that Lee had at first the same.  LCM of 8 and 2 is 8.

1 p + 4 u = 148 - 110
1 p + 4 u = 38
1 p = 38 - 4 u --- (1)

1 p + 4 u + 9 u = 110
1 p + 13 u = 110
1 p = 110 - 13 u --- (2)

(1) = (2)
38 - 4 u = 110 - 13 u
13 u - 4 u = 110 - 38
13 u - 4 u = 72
9 u = 72

1 u = 72 ÷ 9 = 8

Substitute 1 u = 8 into (1).
1 p = 38 - 4 u
1 p = 38 - 4 x 8
1 p = 38 - 32

1 p = 6

Number of erasers that Justin had at first
= 2 p
= 2 x 6
= 12

Answer(s): 12