Carl, Oscar and Cole had a total of 84 marbles. The ratio of Oscar's marbles to Cole's marbles was 8 : 9 at first. Carl and Oscar each gave away ¹₂ of their marbles. Given that the three boys had 60 marbles left, how many marbles did Carl have at first?

	Carl	Oscar	Cole	Total
Comparing Oscar and Cole at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	84
Change	- 1 p	-1x4 = - 4 u		- 24
After	1 p	1x4 = 4 u	9 u	60

Total number of marbles that Carl and Oscar gave away
= 84 - 60
= 24

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

1 p + 4 u = 84 - 60
1 p + 4 u = 24
1 p = 24 - 4 u --- (1)

1 p + 4 u + 9 u = 60
1 p + 13 u = 60
1 p = 60 - 13 u --- (2)

(1) = (2)
24 - 4 u = 60 - 13 u
13 u - 4 u = 60 - 24
13 u - 4 u = 36
9 u = 36

1 u = 36 ÷ 9 = 4

Substitute 1 u = 4 into (1).
1 p = 24 - 4 u
1 p = 24 - 4 x 4
1 p = 24 - 16

1 p = 8

Number of marbles that Carl had at first
= 2 p
= 2 x 8
= 16

Answer(s): 16