²₉ of the people at a concert hall were women. 90% of the remaining people were men and the rest were children. The total number of women and children is 720 less than the number of men. After some men entered the concert hall, 75% of the people that remained were men. How many men entered the concert hall?

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

	Women and children	Men
Before	27x1 = 27 u	63x1 = 63 u
Change		+ 18 u
After	1x27 = 27 u	3x27 = 81 u

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

The total number of children and men 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 men entered, the total number of women and children remains unchanged. Make the total number of women and children the same. LCM of 27 and 1 is 27.

The total number of women and children is 720 less than the number of men.

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

36 u = 720

1 u = 720 ÷ 36 = 20

Number of men who entered the concert hall
= 81 u - 63 u
= 18 u
= 18 x 20
= 360

Answer(s): 360