Pierre, Ahmad and Zeph had a total of 67 marbles. The ratio of Ahmad's marbles to Zeph's marbles was 8 : 9 at first. Pierre and Ahmad each gave away ¹₂ of their marbles. Given that the three boys had 47 marbles left, how many marbles did Pierre have at first?

	Pierre	Ahmad	Zeph	Total
Comparing Ahmad and Zeph at first		8 u	9 u
Before	2 p	2x4 = 8 u	9 u	67
Change	- 1 p	-1x4 = - 4 u		- 20
After	1 p	1x4 = 4 u	9 u	47

Total number of marbles that Pierre and Ahmad gave away
= 67 - 47
= 20

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

1 p + 4 u = 67 - 47
1 p + 4 u = 20
1 p = 20 - 4 u --- (1)

1 p + 4 u + 9 u = 47
1 p + 13 u = 47
1 p = 47 - 13 u --- (2)

(1) = (2)
20 - 4 u = 47 - 13 u
13 u - 4 u = 47 - 20
13 u - 4 u = 27
9 u = 27

1 u = 27 ÷ 9 = 3

Substitute 1 u = 3 into (1).
1 p = 20 - 4 u
1 p = 20 - 4 x 3
1 p = 20 - 12

1 p = 8

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

Answer(s): 16