Cole, George and Charlie had a total of 88 marbles. The ratio of George's marbles to Charlie's marbles was 4 : 9 at first. Cole and George each gave away ¹₂ of their marbles. Given that the three boys had 71 marbles left, how many marbles did Cole have at first?

	Cole	George	Charlie	Total
Comparing George and Charlie at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	88
Change	- 1 p	-1x2 = - 2 u		- 17
After	1 p	1x2 = 2 u	9 u	71

Total number of marbles that Cole and George gave away
= 88 - 71
= 17

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

1 p + 2 u = 88 - 71
1 p + 2 u = 17
1 p = 17 - 2 u --- (1)

1 p + 2 u + 9 u = 71
1 p + 11 u = 71
1 p = 71 - 11 u --- (2)

(1) = (2)
17 - 2 u = 71 - 11 u
11 u - 2 u = 71 - 17
11 u - 2 u = 54
9 u = 54

1 u = 54 ÷ 9 = 6

Substitute 1 u = 6 into (1).
1 p = 17 - 2 u
1 p = 17 - 2 x 6
1 p = 17 - 12

1 p = 5

Number of marbles that Cole had at first
= 2 p
= 2 x 5
= 10

Answer(s): 10