Henry, Luis and Daniel had a total of 69 erasers. The ratio of Luis's erasers to Daniel's erasers was 2 : 5 at first. Henry and Luis each gave away ¹₂ of their erasers. Given that the three boys had 52 erasers left, how many erasers did Henry have in the end?

	Henry	Luis	Daniel	Total
Comparing Luis and Daniel at first		2 u	5 u
Before	2 p	2x1 = 2 u	5 u	69
Change	- 1 p	-1x1 = - 1 u		- 17
After	1 p	1x1 = 1 u	5 u	52

Total number of erasers that Henry and Luis gave away
= 69 - 52
= 17

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

1 p + 1 u = 69 - 52
1 p + 1 u = 17
1 p = 17 - 1 u --- (1)

1 p + 1 u + 5 u = 52
1 p + 6 u = 52
1 p = 52 - 6 u --- (2)

(1) = (2)
17 - 1 u = 52 - 6 u
6 u - 1 u = 52 - 17
6 u - 1 u = 35
5 u = 35

1 u = 35 ÷ 5 = 7

Substitute 1 u = 7 into (1).
1 p = 17 - 1 u
1 p = 17 - 1 x 7
1 p = 17 - 7

1 p = 10

Number of erasers that Henry had in the end
= 1 p
= 10

Answer(s): 10