Tim, Seth and Riordan had a total of 63 marbles. The ratio of Seth's marbles to Riordan's marbles was 8 : 9 at first. Tim and Seth each gave away ¹₂ of their marbles. Given that the three boys had 45 marbles left, how many marbles did Tim have at first?

	Tim	Seth	Riordan	Total
Comparing Seth and Riordan at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	63
Change	- 1 p	-1x4 = - 4 u		- 18
After	1 p	1x4 = 4 u	9 u	45

Total number of marbles that Tim and Seth gave away
= 63 - 45
= 18

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

1 p + 4 u = 63 - 45
1 p + 4 u = 18
1 p = 18 - 4 u --- (1)

1 p + 4 u + 9 u = 45
1 p + 13 u = 45
1 p = 45 - 13 u --- (2)

(1) = (2)
18 - 4 u = 45 - 13 u
13 u - 4 u = 45 - 18
13 u - 4 u = 27
9 u = 27

1 u = 27 ÷ 9 = 3

Substitute 1 u = 3 into (1).
1 p = 18 - 4 u
1 p = 18 - 4 x 3
1 p = 18 - 12

1 p = 6

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

Answer(s): 12