At a barbeque, ⁷₁₀ of the guests were men and the rest were women. Some men then left the barbeque and the remaining number of men were ²₇ of the guests. 82 men then joined the barbeque. The number of guests in the end was 24 more than the number of guests at first. How many guests were there at first?

	Men	Women	Total
Before	7x5 = 35 u	3x5 = 15 u	10x5 = 50 u
Change 1	- ?
After 1	2x3 = 6 u	5x3 = 15 u
Change 2	+ 82
After 2	6 u + 82	15 u	21 u + 82

The number of women remains unchanged.  Make the number of women the same.  LCM of 3 and 5 is 15.

The number of guests in the end was 24 more than the number of guests at first.
If 24 guests are added to the number of guests at first, the number of guests in the end and at first will be the same.

50 u + 24 = 21 u + 82
50 u - 21 u = 82 - 24
29 u = 58

1 u = 58 ÷ 29 = 2

Number of guests at first
= 50 u
= 50 x 2
= 100

Answer(s): 100