¹₉ of the guests at a museum were men. 90% of the remaining guests were women and the rest were children. The number of women was 270 more than the total number of men and children. After some women left the museum, 50% of the guests that remained were women. How many women left the museum?

Men	Children	Women	Total
1x5	8x5		9x5
	1x4	9x4
5	4	36	45

	Men and children	Women
Before	9x1 = 9 u	36x1 = 36 u
Change		- 27 u
After	1x9 = 9 u	1x9 = 9 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 8 and 10 is 40.

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 9 and 1 is 9.

The number of women was 270 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
= 36 u - 9 u
= 27 u

27 u = 270

1 u = 270 ÷ 27 = 10

Number of women who left the museum
= 36 u - 9 u
= 27 u
= 27 x 10
= 270

Answer(s): 270