Ivan, Jeremy and Peter had a total of 73 marbles. The ratio of Jeremy's marbles to Peter's marbles was 4 : 3 at first. Ivan and Jeremy each gave away ¹₂ of their marbles. Given that the three boys had 47 marbles left, how many marbles did Ivan have at first?

	Ivan	Jeremy	Peter	Total
Comparing Jeremy and Peter at first		4 u	3 u
Before	2 p	2x2 = 4 u	3 u	73
Change	- 1 p	-1x2 = - 2 u		- 26
After	1 p	1x2 = 2 u	3 u	47

Total number of marbles that Ivan and Jeremy gave away
= 73 - 47
= 26

The number of marbles that Jeremy had at first is repeated.  Make the number of marbles that Jeremy had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 73 - 47
1 p + 2 u = 26
1 p = 26 - 2 u --- (1)

1 p + 2 u + 3 u = 47
1 p + 5 u = 47
1 p = 47 - 5 u --- (2)

(1) = (2)
26 - 2 u = 47 - 5 u
5 u - 2 u = 47 - 26
5 u - 2 u = 21
3 u = 21

1 u = 21 ÷ 3 = 7

Substitute 1 u = 7 into (1).
1 p = 26 - 2 u
1 p = 26 - 2 x 7
1 p = 26 - 14

1 p = 12

Number of marbles that Ivan had at first
= 2 p
= 2 x 12
= 24

Answer(s): 24