Let's go through the steps to find out how many hours all of the violinists practice.
Step 1: Determine how many members play the violin.
We know there are 168 members in the orchestra and that \(\frac{3}{8}\) of them play the violin.
\[ \text{Number of violinists} = \frac{3}{8} \times 168 \]
Calculating that:
\[ \text{Number of violinists} = \frac{3 \times 168}{8} = \frac{504}{8} = 63 \]
Step 2: Determine how much time each violinist spends practicing each week.
Each non-violin member practices 40 hours each week. First, we need to find the number of non-violin members:
\[ \text{Number of non-violinists} = 168 - 63 = 105 \]
Each non-violin member practices 40 hours, so the total practice time for non-violinists is:
\[ \text{Total hours practiced by non-violinists} = 105 \times 40 = 4200 \text{ hours} \]
Now, we need to find out how much time each violinist practices. The problem states that they practice \(\frac{2}{5}\) of the amount of time that non-violinists practice, which is 40 hours.
Calculating the practice time for each violinist:
\[ \text{Hours per violinist} = \frac{2}{5} \times 40 = \frac{80}{5} = 16 \text{ hours} \]
Step 3: Use the answers from steps 1 & 2 to determine the total time spent practicing by all violinists.
Now, we calculate the total practice time for all the violinists:
\[ \text{Total time practiced by all violinists} = \text{Number of violinists} \times \text{Hours per violinist} \] \[ \text{Total time practiced by all violinists} = 63 \times 16 = 1008 \text{ hours} \]
Summary of Answers
- Number of violinists: 63
- Hours per violinist: 16 hours
- Total time practiced by all violinists: 1008 hours
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