David, Gabriel and Albert had a total of 118 marbles. The ratio of Gabriel's marbles to Albert's marbles was 4 : 7 at first. David and Gabriel each gave away ¹₂ of their marbles. Given that the three boys had 94 marbles left, how many marbles did David have at first?

	David	Gabriel	Albert	Total
Comparing Gabriel and Albert at first		4 u	7 u
Before	2 p	2x2 = 4 u	7 u	118
Change	- 1 p	-1x2 = - 2 u		- 24
After	1 p	1x2 = 2 u	7 u	94

Total number of marbles that David and Gabriel gave away
= 118 - 94
= 24

The number of marbles that Gabriel had at first is repeated.  Make the number of marbles that Gabriel had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 118 - 94
1 p + 2 u = 24
1 p = 24 - 2 u --- (1)

1 p + 2 u + 7 u = 94
1 p + 9 u = 94
1 p = 94 - 9 u --- (2)

(1) = (2)
24 - 2 u = 94 - 9 u
9 u - 2 u = 94 - 24
9 u - 2 u = 70
7 u = 70

1 u = 70 ÷ 7 = 10

Substitute 1 u = 10 into (1).
1 p = 24 - 2 u
1 p = 24 - 2 x 10
1 p = 24 - 20

1 p = 4

Number of marbles that David had at first
= 2 p
= 2 x 4
= 8

Answer(s): 8