At a gathering, ⁴₉ of the guests were women and the rest were men. Some women then left the gathering and the remaining number of women were ²₅ of the guests. 89 women then joined the gathering. The number of guests in the end was 27 more than the number of guests at first. How many guests were there at first?

	Women	Men	Total
Before	4x3 = 12 u	5x3 = 15 u	9x3 = 27 u
Change 1	- ?
After 1	2x5 = 10 u	3x5 = 15 u
Change 2	+ 89
After 2	10 u + 89	15 u	25 u + 89

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

The number of guests in the end was 27 more than the number of guests at first.
If 27 guests are added to the number of guests at first, the number of guests in the end and at first will be the same.

27 u + 27 = 25 u + 89
27 u - 25 u = 89 - 27
2 u = 62

1 u = 62 ÷ 2 = 31

Number of guests at first
= 27 u
= 27 x 31
= 837

Answer(s): 837