¹₉ 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 378 more than the total number of men and children. After some women left the concert hall, 60% of the guests that remained were women. How many women left the concert hall?

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

	Men and children	Women
Before	9x2 = 18 u	36x2 = 72 u
Change		- 45 u
After	2x9 = 18 u	3x9 = 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 8 and 10 is 40.

60% = ⁶⁰₁₀₀ = ³₅

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 2 is 18.

The number of women was 378 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
= 72 u - 18 u
= 54 u

54 u = 378

1 u = 378 ÷ 54 = 7

Number of women who left the concert hall
= 72 u - 27 u
= 45 u
= 45 x 7
= 315

Answer(s): 315