Mark, Asher and Luis had a total of 151 marbles. The ratio of Asher's marbles to Luis's marbles was 8 : 5 at first. Mark and Asher each gave away ¹₂ of their marbles. Given that the three boys had 103 marbles left, how many marbles did Mark have in the end?

	Mark	Asher	Luis	Total
Comparing Asher and Luis at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	151
Change	- 1 p	-1x4 = - 4 u		- 48
After	1 p	1x4 = 4 u	5 u	103

Total number of marbles that Mark and Asher gave away
= 151 - 103
= 48

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

1 p + 4 u = 151 - 103
1 p + 4 u = 48
1 p = 48 - 4 u --- (1)

1 p + 4 u + 5 u = 103
1 p + 9 u = 103
1 p = 103 - 9 u --- (2)

(1) = (2)
48 - 4 u = 103 - 9 u
9 u - 4 u = 103 - 48
9 u - 4 u = 55
5 u = 55

1 u = 55 ÷ 5 = 11

Substitute 1 u = 11 into (1).
1 p = 48 - 4 u
1 p = 48 - 4 x 11
1 p = 48 - 44

1 p = 4

Number of marbles that Mark had in the end
= 1 p
= 4

Answer(s): 4