Eric, Paul and Vincent had a total of 71 marbles. The ratio of Paul's marbles to Vincent's marbles was 4 : 3 at first. Eric and Paul each gave away ¹₂ of their marbles. Given that the three boys had 49 marbles left, how many marbles did Eric have at first?

	Eric	Paul	Vincent	Total
Comparing Paul and Vincent at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	71
Change	- 1 p	-1x2 = - 2 u		- 22
After	1 p	1x2 = 2 u	3 u	49

Total number of marbles that Eric and Paul gave away
= 71 - 49
= 22

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 4 and 2 is 4.

1 p + 2 u = 71 - 49
1 p + 2 u = 22
1 p = 22 - 2 u --- (1)

1 p + 2 u + 3 u = 49
1 p + 5 u = 49
1 p = 49 - 5 u --- (2)

(1) = (2)
22 - 2 u = 49 - 5 u
5 u - 2 u = 49 - 22
5 u - 2 u = 27
3 u = 27

1 u = 27 ÷ 3 = 9

Substitute 1 u = 9 into (1).
1 p = 22 - 2 u
1 p = 22 - 2 x 9
1 p = 22 - 18

1 p = 4

Number of marbles that Eric had at first
= 2 p
= 2 x 4
= 8

Answer(s): 8