Sure, let's go through the problem step by step to determine what fraction of Frank's customers choose the natural finish.
1. Determine the fraction of customers choosing each wood finish:
- Dark walnut: [tex]\(\frac{1}{8}\)[/tex]
- Golden oak: [tex]\(\frac{2}{5}\)[/tex]
- Cherry: [tex]\(\frac{3}{10}\)[/tex]
2. Add these fractions together to find the total fraction of customers who choose either dark walnut, golden oak, or cherry finishes.
Let's convert these fractions to a common denominator for ease of addition. The least common multiple (LCM) of 8, 5, and 10 is 40:
- Dark walnut:
[tex]\[
\frac{1}{8} = \frac{5}{40}
\][/tex]
- Golden oak:
[tex]\[
\frac{2}{5} = \frac{16}{40}
\][/tex]
- Cherry:
[tex]\[
\frac{3}{10} = \frac{12}{40}
\][/tex]
Now, add these fractions:
[tex]\[
\frac{5}{40} + \frac{16}{40} + \frac{12}{40} = \frac{33}{40}
\][/tex]
3. Calculate the fraction of customers who prefer a natural finish.
Since the total fraction of all customers must equal 1 (i.e., 100% of the customers), we subtract the total fraction of the customers choosing dark walnut, golden oak, or cherry from 1:
[tex]\[
1 - \frac{33}{40} = \frac{40}{40} - \frac{33}{40} = \frac{7}{40}
\][/tex]
So, the fraction of Frank's customers who choose the natural finish is [tex]\(\frac{7}{40}\)[/tex]. This can also be written as a decimal, which is approximately 0.175.
