Owen, Paul and David had a total of 51 marbles. The ratio of Paul's marbles to David's marbles was 8 : 5 at first. Owen and Paul each gave away ¹₂ of their marbles. Given that the three boys had 33 marbles left, how many marbles did Owen have at first?

	Owen	Paul	David	Total
Comparing Paul and David at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	51
Change	- 1 p	-1x4 = - 4 u		- 18
After	1 p	1x4 = 4 u	5 u	33

Total number of marbles that Owen and Paul gave away
= 51 - 33
= 18

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

1 p + 4 u = 51 - 33
1 p + 4 u = 18
1 p = 18 - 4 u --- (1)

1 p + 4 u + 5 u = 33
1 p + 9 u = 33
1 p = 33 - 9 u --- (2)

(1) = (2)
18 - 4 u = 33 - 9 u
9 u - 4 u = 33 - 18
9 u - 4 u = 15
5 u = 15

1 u = 15 ÷ 5 = 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 Owen had at first
= 2 p
= 2 x 6
= 12

Answer(s): 12