To determine the ratio of the number of men, children, and women at first, let’s follow the steps to find the initial counts of each group based on the given information.
1. Let [tex]\( V \)[/tex] be the total number of visitors initially.
2. Calculate the number of men:
[tex]\[
\text{Men} = \frac{2}{5} \times V = \frac{2V}{5}
\][/tex]
3. Calculate the remaining visitors after accounting for men:
[tex]\[
\text{Remaining visitors} = V - \frac{2V}{5} = \frac{3V}{5}
\][/tex]
4. Determine the number of children, which is [tex]\(\frac{1}{4}\)[/tex] of the remaining visitors:
[tex]\[
\text{Children} = \frac{1}{4} \times \frac{3V}{5} = \frac{3V}{20}
\][/tex]
5. The number of women initially is the remaining visitors after accounting for both men and children:
[tex]\[
\text{Women} = \frac{3V}{5} - \frac{3V}{20} = \frac{12V}{20} - \frac{3V}{20} = \frac{9V}{20}
\][/tex]
Now we need to establish the initial numbers and their ratio:
6. Determine the initial counts of each group:
[tex]\[
\text{Men} = \frac{2V}{5}
\][/tex]
[tex]\[
\text{Children} = \frac{3V}{20}
\][/tex]
[tex]\[
\text{Women} = \frac{9V}{20}
\][/tex]
7. Assigning the values from the numerical result:
- Total visitors [tex]\( V = 760 \)[/tex]
- Number of men = [tex]\( \frac{2 \times 760}{5} = 304 \)[/tex]
- Number of children = [tex]\( \frac{3 \times 760}{20} = 114 \)[/tex]
- Number of women = [tex]\( \frac{9 \times 760}{20} = 342 \)[/tex]
8. Therefore, the initial ratio of the number of men to the number of children to the number of women is:
[tex]\[
\text{Men : Children : Women} = 304 : 114 : 342
\][/tex]
To simplify:
[tex]\[
304 : 114 : 342 \approx 152 : 57 : 171 \approx 8 : 3 : 9
\][/tex]
Hence, the initial ratio is:
[tex]\[
\boxed{8 : 3 : 9}
\][/tex]
