Tom, Caden and Flynn had a total of 73 erasers. The ratio of Caden's erasers to Flynn's erasers was 2 : 3 at first. Tom and Caden each gave away ¹₂ of their erasers. Given that the three boys had 53 erasers left, how many erasers did Tom have at first?

	Tom	Caden	Flynn	Total
Comparing Caden and Flynn at first		2 u	3 u
Before	2 p	2x1 = 2 u	3 u	73
Change	- 1 p	-1x1 = - 1 u		- 20
After	1 p	1x1 = 1 u	3 u	53

Total number of erasers that Tom and Caden gave away
= 73 - 53
= 20

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

1 p + 1 u = 73 - 53
1 p + 1 u = 20
1 p = 20 - 1 u --- (1)

1 p + 1 u + 3 u = 53
1 p + 4 u = 53
1 p = 53 - 4 u --- (2)

(1) = (2)
20 - 1 u = 53 - 4 u
4 u - 1 u = 53 - 20
4 u - 1 u = 33
3 u = 33

1 u = 33 ÷ 3 = 11

Substitute 1 u = 11 into (1).
1 p = 20 - 1 u
1 p = 20 - 1 x 11
1 p = 20 - 11

1 p = 9

Number of erasers that Tom had at first
= 2 p
= 2 x 9
= 18

Answer(s): 18