Cole and Ian shared some buttons in the ratio of 4 : 7. After both each gave away 117 buttons to their brother, the ratio of the number of buttons Cole had to the total number of buttons left was 1 : 6. How many buttons did the two boys have at first?

	Cole	Ian	Difference
Before	4x4 = 16 u	7x4 = 28 u	3x4 = 12 u
Change	- 117	- 117
After	1x3 = 3 u	5x3 = 15 u	4x3 = 12 u

Number of buttons that Cole and Ian each gave to their brother is the same.

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

Number of buttons that Cole gave to their brother
= 16 u - 3 u
= 13 u

13 u = 117

1 u = 117 ÷ 13 = 9

Number of buttons that the two boys had at first
= 16 u + 28 u
= 44 u
= 44 x 9
= 396

Answer(s): 396