Tim, Zeph and Ahmad had a total of 70 beads. The ratio of Zeph's beads to Ahmad's beads was 2 : 5 at first. Tim and Zeph each gave away ¹₂ of their beads. Given that the three boys had 55 beads left, how many beads did Tim have in the end?

	Tim	Zeph	Ahmad	Total
Comparing Zeph and Ahmad at first		2 u	5 u
Before	2 p	2x1 = 2 u	5 u	70
Change	- 1 p	-1x1 = - 1 u		- 15
After	1 p	1x1 = 1 u	5 u	55

Total number of beads that Tim and Zeph gave away
= 70 - 55
= 15

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

1 p + 1 u = 70 - 55
1 p + 1 u = 15
1 p = 15 - 1 u --- (1)

1 p + 1 u + 5 u = 55
1 p + 6 u = 55
1 p = 55 - 6 u --- (2)

(1) = (2)
15 - 1 u = 55 - 6 u
6 u - 1 u = 55 - 15
6 u - 1 u = 40
5 u = 40

1 u = 40 ÷ 5 = 8

Substitute 1 u = 8 into (1).
1 p = 15 - 1 u
1 p = 15 - 1 x 8
1 p = 15 - 8

1 p = 7

Number of beads that Tim had in the end
= 1 p
= 7

Answer(s): 7