At a school carnival, each man received 5 packets of caramel apples and each woman received 6 packets of caramel apples. Each accompanying child received 8 packets of caramel apples. 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 4 : 7. Given that only 4374 packets of caramel apples were given away, how many people were there at the carnival?

Men	Women	Child
5x11 = 55 u		2x11 = 22 u
4x5 = 20 u	7x5 = 35 u	22 u

	Men	Women	Child
Number	20 u	35 u	22 u
Value	5	6	8
Total Value	100 u	210 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 caramel apples given away
= 20 u x 5 + 35 u x 6 + 22 u x 8
= 100 u + 210 u + 176 u
= 486 u

486 u = 4374

1 u = 4374 ÷ 486 = 9

Number of people at the carnival
= 55 u + 22 u
= 77 u
= 77 x 9
= 693

Answer(s): 693