To determine the fraction of directors who opted for the futuristic theme, we need to start by calculating the fractions of directors who chose the other three themes.
1. Vintage Theme:
Given that [tex]\(\frac{1}{8}\)[/tex] of the directors choose the vintage theme.
2. Modern Loft Theme:
Given that [tex]\(\frac{2}{5}\)[/tex] of the directors choose the modern loft theme.
3. Countryside Theme:
Given that [tex]\(\frac{3}{10}\)[/tex] of the directors choose the countryside setting.
Next, we sum up these fractions to find the total fraction of directors who chose either the vintage, modern loft, or countryside themes.
Vintage [tex]\(= \frac{1}{8} = 0.125\)[/tex]
Modern Loft [tex]\(= \frac{2}{5} = 0.4\)[/tex]
Countryside [tex]\(= \frac{3}{10} = 0.3\)[/tex]
Adding these fractions together:
[tex]\[
0.125 + 0.4 + 0.3 = 0.825
\][/tex]
This sum represents the fraction of directors who did not choose the futuristic theme.
Finally, to find the fraction that represents the directors who chose the futuristic theme, we subtract the sum of the other themes from the total (which is 1 since this represents the whole or 100%):
[tex]\[
1 - 0.825 = 0.175
\][/tex]
To express this result as a fraction:
[tex]\[
0.175 = \frac{175}{1000} = \frac{7}{40}
\][/tex]
Thus, the fraction of directors opting for the futuristic theme is [tex]\(\frac{7}{40}\)[/tex].
The correct choice is:
[tex]\[
\boxed{\frac{7}{40}}
\][/tex]
