At a party, ⁵₇ of the guests were men and the rest were women. Some men then left the party and the remaining number of men were ²₅ of the guests. 71 men then joined the party. The number of guests in the end was 38 more than the number of guests at first. How many guests were there at first?

	Men	Women	Total
Before	5x3 = 15 u	2x3 = 6 u	7x3 = 21 u
Change 1	- ?
After 1	2x2 = 4 u	3x2 = 6 u
Change 2	+ 71
After 2	4 u + 71	6 u	10 u + 71

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

The number of guests in the end was 38 more than the number of guests at first.
If 38 guests are added to the number of guests at first, the number of guests in the end and at first will be the same.

21 u + 38 = 10 u + 71
21 u - 10 u = 71 - 38
11 u = 33

1 u = 33 ÷ 11 = 3

Number of guests at first
= 21 u
= 21 x 3
= 63

Answer(s): 63