Tommy, Luke and Wesley had a total of 91 marbles. The ratio of Luke's marbles to Wesley's marbles was 8 : 9 at first. Tommy and Luke each gave away ¹₂ of their marbles. Given that the three boys had 68 marbles left, how many marbles did Tommy have in the end?

	Tommy	Luke	Wesley	Total
Comparing Luke and Wesley at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	91
Change	- 1 p	-1x4 = - 4 u		- 23
After	1 p	1x4 = 4 u	9 u	68

Total number of marbles that Tommy and Luke gave away
= 91 - 68
= 23

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

1 p + 4 u = 91 - 68
1 p + 4 u = 23
1 p = 23 - 4 u --- (1)

1 p + 4 u + 9 u = 68
1 p + 13 u = 68
1 p = 68 - 13 u --- (2)

(1) = (2)
23 - 4 u = 68 - 13 u
13 u - 4 u = 68 - 23
13 u - 4 u = 45
9 u = 45

1 u = 45 ÷ 9 = 5

Substitute 1 u = 5 into (1).
1 p = 23 - 4 u
1 p = 23 - 4 x 5
1 p = 23 - 20

1 p = 3

Number of marbles that Tommy had in the end
= 1 p
= 3

Answer(s): 3