Dylan, Ethan and Eric had a total of 116 marbles. The ratio of Ethan's marbles to Eric's marbles was 8 : 5 at first. Dylan and Ethan each gave away ¹₂ of their marbles. Given that the three boys had 78 marbles left, how many marbles did Dylan have at first?

	Dylan	Ethan	Eric	Total
Comparing Ethan and Eric at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	116
Change	- 1 p	-1x4 = - 4 u		- 38
After	1 p	1x4 = 4 u	5 u	78

Total number of marbles that Dylan and Ethan gave away
= 116 - 78
= 38

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

1 p + 4 u = 116 - 78
1 p + 4 u = 38
1 p = 38 - 4 u --- (1)

1 p + 4 u + 5 u = 78
1 p + 9 u = 78
1 p = 78 - 9 u --- (2)

(1) = (2)
38 - 4 u = 78 - 9 u
9 u - 4 u = 78 - 38
9 u - 4 u = 40
5 u = 40

1 u = 40 ÷ 5 = 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 marbles that Dylan had at first
= 2 p
= 2 x 6
= 12

Answer(s): 12