Albert, Vincent and Sam had a total of 34 buttons. The ratio of Vincent's buttons to Sam's buttons was 2 : 5 at first. Albert and Vincent each gave away ¹₂ of their buttons. Given that the three boys had 27 buttons left, how many buttons did Albert have at first?

	Albert	Vincent	Sam	Total
Comparing Vincent and Sam at first		2 u	5 u
Before	2 p	2x1 = 2 u	5 u	34
Change	- 1 p	-1x1 = - 1 u		- 7
After	1 p	1x1 = 1 u	5 u	27

Total number of buttons that Albert and Vincent gave away
= 34 - 27
= 7

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

1 p + 1 u = 34 - 27
1 p + 1 u = 7
1 p = 7 - 1 u --- (1)

1 p + 1 u + 5 u = 27
1 p + 6 u = 27
1 p = 27 - 6 u --- (2)

(1) = (2)
7 - 1 u = 27 - 6 u
6 u - 1 u = 27 - 7
6 u - 1 u = 20
5 u = 20

1 u = 20 ÷ 5 = 4

Substitute 1 u = 4 into (1).
1 p = 7 - 1 u
1 p = 7 - 1 x 4
1 p = 7 - 4

1 p = 3

Number of buttons that Albert had at first
= 2 p
= 2 x 3
= 6

Answer(s): 6