Luke, John and Peter had a total of 92 marbles. The ratio of John's marbles to Peter's marbles was 6 : 7 at first. Luke and John each gave away ¹₂ of their marbles. Given that the three boys had 67 marbles left, how many marbles did Luke have at first?

	Luke	John	Peter	Total
Comparing John and Peter at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	92
Change	- 1 p	-1x3 = - 3 u		- 25
After	1 p	1x3 = 3 u	7 u	67

Total number of marbles that Luke and John gave away
= 92 - 67
= 25

The number of marbles that John had at first is repeated.  Make the number of marbles that John had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 92 - 67
1 p + 3 u = 25
1 p = 25 - 3 u --- (1)

1 p + 3 u + 7 u = 67
1 p + 10 u = 67
1 p = 67 - 10 u --- (2)

(1) = (2)
25 - 3 u = 67 - 10 u
10 u - 3 u = 67 - 25
10 u - 3 u = 42
7 u = 42

1 u = 42 ÷ 7 = 6

Substitute 1 u = 6 into (1).
1 p = 25 - 3 u
1 p = 25 - 3 x 6
1 p = 25 - 18

1 p = 7

Number of marbles that Luke had at first
= 2 p
= 2 x 7
= 14

Answer(s): 14