Valen and Simon shared some buttons in the ratio of 5 : 8. After both each gave away 51 buttons to their cousin, the ratio of the number of buttons Valen had to the total number of buttons left was 1 : 6. How many buttons did the two boys have at first?

	Valen	Simon	Difference
Before	5x4 = 20 u	8x4 = 32 u	3x4 = 12 u
Change	- 51	- 51
After	1x3 = 3 u	5x3 = 15 u	4x3 = 12 u

Number of buttons that Valen and Simon each gave to their cousin is the same.

The difference in the number of buttons at first and in the end between Valen and Simon remains unchanged.  Make the difference in the number of buttons the same.  LCM of 3 and 4 is 12.

Number of buttons that Valen gave to their cousin
= 20 u - 3 u
= 17 u

17 u = 51

1 u = 51 ÷ 17 = 3

Number of buttons that the two boys had at first
= 20 u + 32 u
= 52 u
= 52 x 3
= 156

Answer(s): 156