Shannon, Barbara and Dana had a total of 112 buttons. The ratio of Barbara's buttons to Dana's buttons was at first 4 : 5. After Shannon and Barbara each gave half of their buttons away, the 3 children had 76 buttons left. How many buttons did Shannon and Barbara have at first?

	Shannon	Barbara	Dana	Total
Before	2 p	4 u	5 u	112
Change	- 1 p	- 2 u		- 36
After	1 p	2 u	5 u	76

Number of buttons that Shannon and Barbara gave away
= 112 - 76
= 36

1 p + 2 u = 36
1 p = 36 - 2 u --- (1)

1 p + 2 u + 5 u = 76
1 p + 7 u = 76
1 p = 76 - 7 u --- (2)

(1) = (2)
36 - 2 u = 76 - 7 u
7 u - 2 u = 76 - 36
5 u = 40

1 u = 40 ÷ 5 = 8

From (1)
1 p = 36 - 2 u
1 p = 36 - 2 x 8

1 p = 36 - 16 = 20

Number of buttons that Shannon and Barbara had at first
= 2 p + 4 u
= 2 x 20 + 4 x 8
= 40 + 40
= 80

Answer(s): 80