Charlie, Jack and Japheth had a total of 96 coins. The ratio of Jack's coins to Japheth's coins was 2 : 9 at first. Charlie and Jack each gave away ¹₂ of their coins. Given that the three boys had 84 coins left, how many coins did Charlie have at first?

	Charlie	Jack	Japheth	Total
Comparing Jack and Japheth at first		2 u	9 u
Before	2 p	2x1 = 2 u	9 u	96
Change	- 1 p	-1x1 = - 1 u		- 12
After	1 p	1x1 = 1 u	9 u	84

Total number of coins that Charlie and Jack gave away
= 96 - 84
= 12

The number of coins that Jack had at first is repeated.  Make the number of coins that Jack had at first the same.  LCM of 2 and 2 is 2.

1 p + 1 u = 96 - 84
1 p + 1 u = 12
1 p = 12 - 1 u --- (1)

1 p + 1 u + 9 u = 84
1 p + 10 u = 84
1 p = 84 - 10 u --- (2)

(1) = (2)
12 - 1 u = 84 - 10 u
10 u - 1 u = 84 - 12
10 u - 1 u = 72
9 u = 72

1 u = 72 ÷ 9 = 8

Substitute 1 u = 8 into (1).
1 p = 12 - 1 u
1 p = 12 - 1 x 8
1 p = 12 - 8

1 p = 4

Number of coins that Charlie had at first
= 2 p
= 2 x 4
= 8

Answer(s): 8