Ethan, Julian and Ivan had a total of 184 marbles. The ratio of Julian's marbles to Ivan's marbles was 8 : 9 at first. Ethan and Julian each gave away ¹₂ of their marbles. Given that the three boys had 137 marbles left, how many marbles did Ethan have at first?

	Ethan	Julian	Ivan	Total
Comparing Julian and Ivan at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	184
Change	- 1 p	-1x4 = - 4 u		- 47
After	1 p	1x4 = 4 u	9 u	137

Total number of marbles that Ethan and Julian gave away
= 184 - 137
= 47

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

1 p + 4 u = 184 - 137
1 p + 4 u = 47
1 p = 47 - 4 u --- (1)

1 p + 4 u + 9 u = 137
1 p + 13 u = 137
1 p = 137 - 13 u --- (2)

(1) = (2)
47 - 4 u = 137 - 13 u
13 u - 4 u = 137 - 47
13 u - 4 u = 90
9 u = 90

1 u = 90 ÷ 9 = 10

Substitute 1 u = 10 into (1).
1 p = 47 - 4 u
1 p = 47 - 4 x 10
1 p = 47 - 40

1 p = 7

Number of marbles that Ethan had at first
= 2 p
= 2 x 7
= 14

Answer(s): 14