Harry, Archie and Ahmad had a total of 85 erasers. The ratio of Archie's erasers to Ahmad's erasers was 2 : 5 at first. Harry and Archie each gave away ¹₂ of their erasers. Given that the three boys had 65 erasers left, how many erasers did Harry have at first?

	Harry	Archie	Ahmad	Total
Comparing Archie and Ahmad at first		2 u	5 u
Before	2 p	2x1 = 2 u	5 u	85
Change	- 1 p	-1x1 = - 1 u		- 20
After	1 p	1x1 = 1 u	5 u	65

Total number of erasers that Harry and Archie gave away
= 85 - 65
= 20

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

1 p + 1 u = 85 - 65
1 p + 1 u = 20
1 p = 20 - 1 u --- (1)

1 p + 1 u + 5 u = 65
1 p + 6 u = 65
1 p = 65 - 6 u --- (2)

(1) = (2)
20 - 1 u = 65 - 6 u
6 u - 1 u = 65 - 20
6 u - 1 u = 45
5 u = 45

1 u = 45 ÷ 5 = 9

Substitute 1 u = 9 into (1).
1 p = 20 - 1 u
1 p = 20 - 1 x 9
1 p = 20 - 9

1 p = 11

Number of erasers that Harry had at first
= 2 p
= 2 x 11
= 22

Answer(s): 22