Rael, Ben and Bobby had a total of 124 marbles. The ratio of Ben's marbles to Bobby's marbles was 8 : 7 at first. Rael and Ben each gave away ¹₂ of their marbles. Given that the three boys had 90 marbles left, how many marbles did Rael have at first?

	Rael	Ben	Bobby	Total
Comparing Ben and Bobby at first		8 u	7 u
Before	2 p	2x4 = 8 u	7 u	124
Change	- 1 p	-1x4 = - 4 u		- 34
After	1 p	1x4 = 4 u	7 u	90

Total number of marbles that Rael and Ben gave away
= 124 - 90
= 34

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

1 p + 4 u = 124 - 90
1 p + 4 u = 34
1 p = 34 - 4 u --- (1)

1 p + 4 u + 7 u = 90
1 p + 11 u = 90
1 p = 90 - 11 u --- (2)

(1) = (2)
34 - 4 u = 90 - 11 u
11 u - 4 u = 90 - 34
11 u - 4 u = 56
7 u = 56

1 u = 56 ÷ 7 = 8

Substitute 1 u = 8 into (1).
1 p = 34 - 4 u
1 p = 34 - 4 x 8
1 p = 34 - 32

1 p = 2

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

Answer(s): 4