At a school carnival, each woman received 5 packets of mochi balls and each man received 7 packets of mochi balls. Each accompanying child received 8 packets of mochi balls. ²₇ of the people at the school carnival were child. The ratio of the number of women to the number of men at the carnival was 3 : 7. Given that only 864 packets of mochi balls were given away, how many people were there at the carnival?

Women	Men	Child
5x2 = 10 u		2x2 = 4 u
3x1 = 3 u	7x1 = 7 u	4 u

	Women	Men	Child
Number	3 u	7 u	4 u
Value	5	7	8
Total Value	15 u	49 u	32 u

Fraction of the people who were adults
= 1 - ²₇
= ⁵₇

The number of adults is the combined repeated identity. Make the number of adults the same.  LCM of 10 and 5 is 10. 

Number of packets of mochi balls given away
= 3 u x 5 + 7 u x 7 + 4 u x 8
= 15 u + 49 u + 32 u
= 96 u

96 u = 864

1 u = 864 ÷ 96 = 9

Number of people at the carnival
= 10 u + 4 u
= 14 u
= 14 x 9
= 126

Answer(s): 126