Fabian, Oscar and Daniel had a total of 129 marbles. The ratio of Oscar's marbles to Daniel's marbles was 6 : 7 at first. Fabian and Oscar each gave away ¹₂ of their marbles. Given that the three boys had 96 marbles left, how many marbles did Fabian have at first?

	Fabian	Oscar	Daniel	Total
Comparing Oscar and Daniel at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	129
Change	- 1 p	-1x3 = - 3 u		- 33
After	1 p	1x3 = 3 u	7 u	96

Total number of marbles that Fabian and Oscar gave away
= 129 - 96
= 33

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 6 and 2 is 6.

1 p + 3 u = 129 - 96
1 p + 3 u = 33
1 p = 33 - 3 u --- (1)

1 p + 3 u + 7 u = 96
1 p + 10 u = 96
1 p = 96 - 10 u --- (2)

(1) = (2)
33 - 3 u = 96 - 10 u
10 u - 3 u = 96 - 33
10 u - 3 u = 63
7 u = 63

1 u = 63 ÷ 7 = 9

Substitute 1 u = 9 into (1).
1 p = 33 - 3 u
1 p = 33 - 3 x 9
1 p = 33 - 27

1 p = 6

Number of marbles that Fabian had at first
= 2 p
= 2 x 6
= 12

Answer(s): 12