At first, Ian, Gwen and John had some mochi balls in the ratio 7 : 5 : 6 respectively. Ian had 406 mochi balls. Then he gave all his mochi balls to Gwen and John. After receiving the mochi balls from Ian, Gwen and John had the same number of mochi balls. How many mochi balls did Gwen receive from Ian?

	Ian	Gwen	John	Total
Before	7x1 = 7 u	5x1 = 5 u	6x1 = 6 u	18x1 = 18 u
Change	- 7 u	+ 7 u
After	0	1x9 = 9 u	1x9 = 9 u	2x9 = 18 u

The total number of mochi balls remains unchanged. Make the total number of mochi balls the same.  LCM of 2 and 18 is 18.

Total number of mochi balls = 18 u

Number of mochi balls that Gwen had in the end 
= 18 u ÷ 2
= 9 u

Number of mochi balls that Ian had at first = 7 u 

7 u = 406

1 u = 406 ÷ 7 = 58

Number of mochi balls that Gwen received from Ian
= 9 u - 5 u
= 4 u
= 4 x 58
= 232

Answer(s): 232