1/6 of the women and 1/4 of the men remained.

Let total number of men be 4 units.
Then, number of men left behind = 1 unit

Based on the question info, we can say that :

Number of people making up 1/6 of women = number of people making up 1/4 of men + 25


So 1/6 of women = 1 unit + 25

Total number of women
= 6 x (1 unit + 25)
= 6 units + 150

Since there are 310 more women than men,

6 units + 150 - 4 units = 310
2 units + 150 = 310
2 units = 310 - 150
2 units = 160
1 unit = 160 ÷ 2 = 80

Number of women at the musical theatre at first

= 6 units + 150
= 6 x 80 + 150
= 480 + 150
= 630

Alternative method :


Find the common denominator of 4 and 6 first.

We can use 12 .

(since 4 x 3 = 12 and 2 x 6 = 12)

Fraction of men remaining = 1/4 = 3/12
Fraction of women remaining = 1/6 = 2/12


If we let number of men at first be 12 units,

Then number of women at first = 12 units + 310

Number of women remaining
= 1/6 x (12 units + 310)
= 2 units + 51⅔


Since there are 25 more women than men remaining,


2 units + 51⅔ - 3 units = 25

51⅔ units - 1 unit = 25

1 unit = 51⅔ units - 25 = 26⅔ units

Total number of men at first
= 12 x 26⅔
= 320

Total number of women at first
= 320 + 310
= 630