²₉ of the guests at a concert hall were men. 90% of the remaining guests were women and the rest were children. The number of women was 324 more than the total number of men and children. After some women left the concert hall, 50% of the guests that remained were women. How many women left the concert hall?

Men	Children	Women	Total
2x10	7x10		9x10
	1x7	9x7
20	7	63	90

	Men and children	Women
Before	27x1 = 27 u	63x1 = 63 u
Change		- 36 u
After	1x27 = 27 u	1x27 = 27 u

90% = ⁹⁰₁₀₀ = ⁹₁₀
Children : Women = 1 : 9

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.

50% = ⁵⁰₁₀₀ = ¹₂

When some women left, the total number of men and children remains unchanged. Make the total number of men and children the same. LCM of 27 and 1 is 27.

The number of women was 324 more than the total number of men and children at first.

Number of more women than the total number of men and children at first
= 63 u - 27 u
= 36 u

36 u = 324

1 u = 324 ÷ 36 = 9

Number of women who left the concert hall
= 63 u - 27 u
= 36 u
= 36 x 9
= 324

Answer(s): 324