At a company event, ⁷₁₀ of the guests were women and the rest were men. Some women then left the company event and the remaining number of women were ³₅ of the guests. 52 women then joined the company event. The number of guests in the end was 32 more than the number of guests at first. How many guests were there at first?

	Women	Men	Total
Before	7x2 = 14 u	3x2 = 6 u	10x2 = 20 u
Change 1	- ?
After 1	3x3 = 9 u	2x3 = 6 u
Change 2	+ 52
After 2	9 u + 52	6 u	15 u + 52

The number of men remains unchanged.  Make the number of men the same.  LCM of 3 and 2 is 6.

The number of guests in the end was 32 more than the number of guests at first.
If 32 guests are added to the number of guests at first, the number of guests in the end and at first will be the same.

20 u + 32 = 15 u + 52
20 u - 15 u = 52 - 32
5 u = 20

1 u = 20 ÷ 5 = 4

Number of guests at first
= 20 u
= 20 x 4
= 80

Answer(s): 80