Neave, Gabriel and Ben had a total of 111 buttons. The ratio of Gabriel's buttons to Ben's buttons was 4 : 9 at first. Neave and Gabriel each gave away ¹₂ of their buttons. Given that the three boys had 87 buttons left, how many buttons did Neave have at first?

	Neave	Gabriel	Ben	Total
Comparing Gabriel and Ben at first		4 u	9 u
Before	2 p	2x2 = 4 u	9 u	111
Change	- 1 p	-1x2 = - 2 u		- 24
After	1 p	1x2 = 2 u	9 u	87

Total number of buttons that Neave and Gabriel gave away
= 111 - 87
= 24

The number of buttons that Gabriel had at first is repeated.  Make the number of buttons that Gabriel had at first the same.  LCM of 4 and 2 is 4.

1 p + 2 u = 111 - 87
1 p + 2 u = 24
1 p = 24 - 2 u --- (1)

1 p + 2 u + 9 u = 87
1 p + 11 u = 87
1 p = 87 - 11 u --- (2)

(1) = (2)
24 - 2 u = 87 - 11 u
11 u - 2 u = 87 - 24
11 u - 2 u = 63
9 u = 63

1 u = 63 ÷ 9 = 7

Substitute 1 u = 7 into (1).
1 p = 24 - 2 u
1 p = 24 - 2 x 7
1 p = 24 - 14

1 p = 10

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

Answer(s): 20