Jeremy, John and Charlie had a total of 42 marbles. The ratio of John's marbles to Charlie's marbles was 4 : 9 at first. Jeremy and John each gave away ¹₂ of their marbles. Given that the three boys had 30 marbles left, how many marbles did Jeremy have at first?

	Jeremy	John	Charlie	Total
Comparing John and Charlie at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	42
Change	- 1 p	-1x2 = - 2 u		- 12
After	1 p	1x2 = 2 u	9 u	30

Total number of marbles that Jeremy and John gave away
= 42 - 30
= 12

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 4 and 2 is 4.

1 p + 2 u = 42 - 30
1 p + 2 u = 12
1 p = 12 - 2 u --- (1)

1 p + 2 u + 9 u = 30
1 p + 11 u = 30
1 p = 30 - 11 u --- (2)

(1) = (2)
12 - 2 u = 30 - 11 u
11 u - 2 u = 30 - 12
11 u - 2 u = 18
9 u = 18

1 u = 18 ÷ 9 = 2

Substitute 1 u = 2 into (1).
1 p = 12 - 2 u
1 p = 12 - 2 x 2
1 p = 12 - 4

1 p = 8

Number of marbles that Jeremy had at first
= 2 p
= 2 x 8
= 16

Answer(s): 16