¹₈ of the guests at a concert hall were men. 70% of the remaining guests were women and the rest were children. The total number of men and children is 216 less than the number of women. After some women entered the concert hall, 75% of the guests that remained were women. How many women entered the concert hall?

Men	Children	Women	Total
1x10	7x10		8x10
	3x7	7x7
10	21	49	80

	Men and children	Women
Before	31x1 = 31 u	49x1 = 49 u
Change		+ 44 u
After	1x31 = 31 u	3x31 = 93 u

70% = ⁷⁰₁₀₀ = ⁷₁₀
Children : Women = 3 : 7

The total number of children and women at first is repeated.  Make the total number of children at first the same. LCM of 7 and 10 is 70.

75% = ⁷⁵₁₀₀ = ³₄

When some women entered, the total number of men and children remains unchanged. Make the total number of men and children the same. LCM of 31 and 1 is 31.

The total number of men and children is 216 less than the number of women.

Number of less men and children than the number of women at first
= 49 u - 31 u
= 18 u

18 u = 216

1 u = 216 ÷ 18 = 12

Number of women who entered the concert hall
= 93 u - 49 u
= 44 u
= 44 x 12
= 528

Answer(s): 528