At a school carnival, each man received 5 packets of mochi balls and each woman 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 men to the number of women at the carnival was 3 : 8. Given that only 5310 packets of mochi balls were given away, how many people were there at the carnival?

Men	Women	Child
5x11 = 55 u		2x11 = 22 u
3x5 = 15 u	8x5 = 40 u	22 u

	Men	Women	Child
Number	15 u	40 u	22 u
Value	5	7	8
Total Value	75 u	280 u	176 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 11 and 5 is 55. 

Number of packets of mochi balls given away
= 15 u x 5 + 40 u x 7 + 22 u x 8
= 75 u + 280 u + 176 u
= 531 u

531 u = 5310

1 u = 5310 ÷ 531 = 10

Number of people at the carnival
= 55 u + 22 u
= 77 u
= 77 x 10
= 770

Answer(s): 770