Jenson, Jeremy and Ken had a total of 108 buttons. The ratio of Jeremy's buttons to Ken's buttons was 8 : 3 at first. Jenson and Jeremy each gave away ¹₂ of their buttons. Given that the three boys had 66 buttons left, how many buttons did Jenson have in the end?

	Jenson	Jeremy	Ken	Total
Comparing Jeremy and Ken at first		8 u	3 u
Before	2 p	2x4 = 8 u	3 u	108
Change	- 1 p	-1x4 = - 4 u		- 42
After	1 p	1x4 = 4 u	3 u	66

Total number of buttons that Jenson and Jeremy gave away
= 108 - 66
= 42

The number of buttons that Jeremy had at first is repeated.  Make the number of buttons that Jeremy had at first the same.  LCM of 8 and 2 is 8.

1 p + 4 u = 108 - 66
1 p + 4 u = 42
1 p = 42 - 4 u --- (1)

1 p + 4 u + 3 u = 66
1 p + 7 u = 66
1 p = 66 - 7 u --- (2)

(1) = (2)
42 - 4 u = 66 - 7 u
7 u - 4 u = 66 - 42
7 u - 4 u = 24
3 u = 24

1 u = 24 ÷ 3 = 8

Substitute 1 u = 8 into (1).
1 p = 42 - 4 u
1 p = 42 - 4 x 8
1 p = 42 - 32

1 p = 10

Number of buttons that Jenson had in the end
= 1 p
= 10

Answer(s): 10