Fred, Ryan and Mark had a total of 110 marbles. The ratio of Ryan's marbles to Mark's marbles was 2 : 9 at first. Fred and Ryan each gave away ¹₂ of their marbles. Given that the three boys had 91 marbles left, how many marbles did Fred have in the end?

	Fred	Ryan	Mark	Total
Comparing Ryan and Mark at first		2 u	9 u
Before	2 p	2x1 = 2 u	9 u	110
Change	- 1 p	-1x1 = - 1 u		- 19
After	1 p	1x1 = 1 u	9 u	91

Total number of marbles that Fred and Ryan gave away
= 110 - 91
= 19

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

1 p + 1 u = 110 - 91
1 p + 1 u = 19
1 p = 19 - 1 u --- (1)

1 p + 1 u + 9 u = 91
1 p + 10 u = 91
1 p = 91 - 10 u --- (2)

(1) = (2)
19 - 1 u = 91 - 10 u
10 u - 1 u = 91 - 19
10 u - 1 u = 72
9 u = 72

1 u = 72 ÷ 9 = 8

Substitute 1 u = 8 into (1).
1 p = 19 - 1 u
1 p = 19 - 1 x 8
1 p = 19 - 8

1 p = 11

Number of marbles that Fred had in the end
= 1 p
= 11

Answer(s): 11