Michael and Caden shared some buttons in the ratio of 5 : 8. After both each gave away 119 buttons to their sister, the ratio of the number of buttons Michael had to the total number of buttons left was 1 : 6. How many buttons did the two boys have at first?

	Michael	Caden	Difference
Before	5x4 = 20 u	8x4 = 32 u	3x4 = 12 u
Change	- 119	- 119
After	1x3 = 3 u	5x3 = 15 u	4x3 = 12 u

Number of buttons that Michael and Caden each gave to their sister is the same.

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

Number of buttons that Michael gave to their sister
= 20 u - 3 u
= 17 u

17 u = 119

1 u = 119 ÷ 17 = 7

Number of buttons that the two boys had at first
= 20 u + 32 u
= 52 u
= 52 x 7
= 364

Answer(s): 364