Elijah, Asher and Zeph had a total of 163 buttons. The ratio of Asher's buttons to Zeph's buttons was 6 : 7 at first. Elijah and Asher each gave away ¹₂ of their buttons. Given that the three boys had 120 buttons left, how many buttons did Elijah have at first?

	Elijah	Asher	Zeph	Total
Comparing Asher and Zeph at first		6 u	7 u
Before	2 p	2x3 = 6 u	7 u	163
Change	- 1 p	-1x3 = - 3 u		- 43
After	1 p	1x3 = 3 u	7 u	120

Total number of buttons that Elijah and Asher gave away
= 163 - 120
= 43

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

1 p + 3 u = 163 - 120
1 p + 3 u = 43
1 p = 43 - 3 u --- (1)

1 p + 3 u + 7 u = 120
1 p + 10 u = 120
1 p = 120 - 10 u --- (2)

(1) = (2)
43 - 3 u = 120 - 10 u
10 u - 3 u = 120 - 43
10 u - 3 u = 77
7 u = 77

1 u = 77 ÷ 7 = 11

Substitute 1 u = 11 into (1).
1 p = 43 - 3 u
1 p = 43 - 3 x 11
1 p = 43 - 33

1 p = 10

Number of buttons that Elijah had at first
= 2 p
= 2 x 10
= 20

Answer(s): 20