To determine the fraction of directors that opt for the futuristic theme, we need to start by summing the fractions of directors who choose the other three themes and then subtract that sum from 1.
1. First, we need to add up the fractions of directors who choose the vintage theme, modern loft, and countryside setting.
Given:
- Vintage theme: [tex]\(\frac{1}{8}\)[/tex]
- Modern loft: [tex]\(\frac{2}{5}\)[/tex]
- Countryside setting: [tex]\(\frac{3}{10}\)[/tex]
2. Let's convert these fractions to a common denominator so that we can comfortably add them together. The least common multiple (LCM) of 8, 5, and 10 is 40.
- Convert [tex]\(\frac{1}{8}\)[/tex] to a fraction over 40:
[tex]\[
\frac{1}{8} = \frac{1 \times 5}{8 \times 5} = \frac{5}{40}
\][/tex]
- Convert [tex]\(\frac{2}{5}\)[/tex] to a fraction over 40:
[tex]\[
\frac{2}{5} = \frac{2 \times 8}{5 \times 8} = \frac{16}{40}
\][/tex]
- Convert [tex]\(\frac{3}{10}\)[/tex] to a fraction over 40:
[tex]\[
\frac{3}{10} = \frac{3 \times 4}{10 \times 4} = \frac{12}{40}
\][/tex]
3. Now, let's add these fractions together:
[tex]\[
\frac{5}{40} + \frac{16}{40} + \frac{12}{40} = \frac{5 + 16 + 12}{40} = \frac{33}{40}
\][/tex]
4. The total fraction of directors who chose the vintage theme, modern loft, and countryside setting is [tex]\(\frac{33}{40}\)[/tex].
5. To find the fraction of directors who opt for the futuristic theme, subtract this sum from 1 (which can be expressed as [tex]\(\frac{40}{40}\)[/tex]):
[tex]\[
1 - \frac{33}{40} = \frac{40}{40} - \frac{33}{40} = \frac{40 - 33}{40} = \frac{7}{40}
\][/tex]
Therefore, the fraction of directors who opt for the futuristic theme is [tex]\(\frac{7}{40}\)[/tex].
Thus, the correct answer is: [tex]\(\frac{7}{40}\)[/tex].
