Tim, Fred and Reggie had a total of 40 beads. The ratio of Fred's beads to Reggie's beads was 6 : 7 at first. Tim and Fred each gave away ¹₂ of their beads. Given that the three boys had 27 beads left, how many beads did Tim have at first?

	Tim	Fred	Reggie	Total
Comparing Fred and Reggie at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	40
Change	- 1 p	-1x3 = - 3 u		- 13
After	1 p	1x3 = 3 u	7 u	27

Total number of beads that Tim and Fred gave away
= 40 - 27
= 13

The number of beads that Fred had at first is repeated.  Make the number of beads that Fred had at first the same.  LCM of 6 and 2 is 6.

1 p + 3 u = 40 - 27
1 p + 3 u = 13
1 p = 13 - 3 u --- (1)

1 p + 3 u + 7 u = 27
1 p + 10 u = 27
1 p = 27 - 10 u --- (2)

(1) = (2)
13 - 3 u = 27 - 10 u
10 u - 3 u = 27 - 13
10 u - 3 u = 14
7 u = 14

1 u = 14 ÷ 7 = 2

Substitute 1 u = 2 into (1).
1 p = 13 - 3 u
1 p = 13 - 3 x 2
1 p = 13 - 6

1 p = 7

Number of beads that Tim had at first
= 2 p
= 2 x 7
= 14

Answer(s): 14