¹₉ of the visitors at a museum were women. 70% of the remaining visitors were men and the rest were children. The total number of women and children is 165 less than the number of men. After some men entered the museum, 80% of the visitors that remained were men. How many men entered the museum?

Women	Children	Men	Total
1x5	8x5		9x5
	3x4	7x4
5	12	28	45

	Women and children	Men
Before	17x1 = 17 u	28x1 = 28 u
Change		+ 40 u
After	1x17 = 17 u	4x17 = 68 u

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

The total number of children and men at first is repeated.  Make the total number of children at first the same. LCM of 8 and 10 is 40.

80% = ⁸⁰₁₀₀ = ⁴₅

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 17 and 1 is 17.

The total number of women and children is 165 less than the number of men.

Number of less women and children than the number of men at first
= 28 u - 17 u
= 11 u

11 u = 165

1 u = 165 ÷ 11 = 15

Number of men who entered the museum
= 68 u - 28 u
= 40 u
= 40 x 15
= 600

Answer(s): 600