At a school carnival, each man received 4 packets of sweets and each woman received 6 packets of sweets. Each accompanying child received 7 packets of sweets. 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 2 : 5. Given that only 2592 packets of sweets were given away, how many people were there at the carnival?

Men	Women	Child
5x7 = 35 u		2x7 = 14 u
2x5 = 10 u	5x5 = 25 u	14 u

	Men	Women	Child
Number	10 u	25 u	14 u
Value	4	6	7
Total Value	40 u	150 u	98 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 7 and 5 is 35. 

Number of packets of sweets given away
= 10 u x 4 + 25 u x 6 + 14 u x 7
= 40 u + 150 u + 98 u
= 288 u

288 u = 2592

1 u = 2592 ÷ 288 = 9

Number of people at the carnival
= 35 u + 14 u
= 49 u
= 49 x 9
= 441

Answer(s): 441