Daniel, George and Cole had a total of 70 beads. The ratio of George's beads to Cole's beads was 8 : 5 at first. Daniel and George each gave away ¹₂ of their beads. Given that the three boys had 45 beads left, how many beads did Daniel have at first?

	Daniel	George	Cole	Total
Comparing George and Cole at first		8 u	5 u
Before	2 p	2x4 = 8 u	5 u	70
Change	- 1 p	-1x4 = - 4 u		- 25
After	1 p	1x4 = 4 u	5 u	45

Total number of beads that Daniel and George gave away
= 70 - 45
= 25

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

1 p + 4 u = 70 - 45
1 p + 4 u = 25
1 p = 25 - 4 u --- (1)

1 p + 4 u + 5 u = 45
1 p + 9 u = 45
1 p = 45 - 9 u --- (2)

(1) = (2)
25 - 4 u = 45 - 9 u
9 u - 4 u = 45 - 25
9 u - 4 u = 20
5 u = 20

1 u = 20 ÷ 5 = 4

Substitute 1 u = 4 into (1).
1 p = 25 - 4 u
1 p = 25 - 4 x 4
1 p = 25 - 16

1 p = 9

Number of beads that Daniel had at first
= 2 p
= 2 x 9
= 18

Answer(s): 18