Fiona, Raeann and Tina had a total of 180 buttons. The ratio of Raeann's buttons to Tina's buttons was at first 12 : 5. After Fiona and Raeann each gave half of their buttons away, the 3 children had 110 buttons left. How many buttons did Fiona and Raeann have at first?

	Fiona	Raeann	Tina	Total
Before	2 p	12 u	5 u	180
Change	- 1 p	- 6 u		- 70
After	1 p	6 u	5 u	110

Number of buttons that Fiona and Raeann gave away
= 180 - 110
= 70

1 p + 6 u = 70
1 p = 70 - 6 u --- (1)

1 p + 6 u + 5 u = 110
1 p + 11 u = 110
1 p = 110 - 11 u --- (2)

(1) = (2)
70 - 6 u = 110 - 11 u
11 u - 6 u = 110 - 70
5 u = 40

1 u = 40 ÷ 5 = 8

From (1)
1 p = 70 - 6 u
1 p = 70 - 6 x 8

1 p = 70 - 48 = 22

Number of buttons that Fiona and Raeann had at first
= 2 p + 12 u
= 2 x 22 + 12 x 8
= 44 + 40
= 84

Answer(s): 84