Let's solve the given problem step-by-step.
Given:
1. [tex]\(\frac{2}{5}\)[/tex] of the visitors in the exhibition hall were men.
2. [tex]\(\frac{1}{4}\)[/tex] of the remaining visitors were children and the rest were women.
3. Another 190 women entered, and the ratio of men to women became [tex]\(4:7\)[/tex].
We need to find:
(a) The ratio of the number of men to the number of children to the number of women at first.
Let's denote the total number of visitors by [tex]\(T\)[/tex].
1. Calculating the number of men (M):
- The number of men in the exhibition hall is given by:
[tex]\[
M = \frac{2}{5}T
\][/tex]
2. Calculating the remaining visitors:
- After accounting for the men, the remaining visitors are:
[tex]\[
T - M = T - \frac{2}{5}T = \frac{3}{5}T
\][/tex]
3. Calculating the number of children (C):
- [tex]\(\frac{1}{4}\)[/tex] of the remaining visitors were children:
[tex]\[
C = \frac{1}{4} \left( \frac{3}{5}T \right) = \frac{3}{20}T
\][/tex]
4. Calculating the number of women (W) initially:
- The rest of the remaining visitors were women:
[tex]\[
W = \frac{3}{5}T - C = \frac{3}{5}T - \frac{3}{20}T = \frac{12}{20}T - \frac{3}{20}T = \frac{9}{20}T
\][/tex]
5. Determining the ratio of men to children to women:
We now need to express the ratio [tex]\(M : C : W\)[/tex].
Substituting the values we found:
[tex]\[
\frac{M}{T} = \frac{2}{5},\quad \frac{C}{T} = \frac{3}{20},\quad \frac{W}{T} = \frac{9}{20}
\][/tex]
Therefore, the ratio is:
[tex]\[
M : C : W = \frac{2}{5}T : \frac{3}{20}T : \frac{9}{20}T
\][/tex]
To simplify, find a common denominator for these fractions:
[tex]\[
M : C : W = \frac{8}{20}T : \frac{3}{20}T : \frac{9}{20}T
\][/tex]
We can cancel out the common factor of [tex]\(\frac{T}{20}\)[/tex]:
[tex]\[
M : C : W = 8 : 3 : 9
\][/tex]
Thus, the ratio of the number of men to the number of children to the number of women at first is [tex]\(8 : 3 : 9\)[/tex].
