To determine how much wire the electrician has left, we need to follow these steps:
- First, convert all mixed numbers and fractions to improper fractions.
- Subtract the lengths of wire used from the total length.
Step 1: Convert to improper fractions
The total amount of wire: \[ 10 \frac{1}{2} = \frac{21}{2} \]
The length of wire cut for the first job: \[ 2 \frac{9}{16} = \frac{2 \times 16 + 9}{16} = \frac{32 + 9}{16} = \frac{41}{16} \]
The length of wire cut for the second job: \[ 4 \frac{3}{8} = \frac{4 \times 8 + 3}{8} = \frac{32 + 3}{8} = \frac{35}{8} \]
Step 2: Convert all fractions to a common denominator
The denominators of the fractions are 2, 16, and 8. The least common multiple of these denominators is 16.
Now we convert each fraction to have a denominator of 16:
- The total wire: \[ \frac{21}{2} = \frac{21 \times 8}{2 \times 8} = \frac{168}{16} \]
- The first job: \[ \frac{41}{16} \quad \text{(no change needed)} \]
- The second job: \[ \frac{35}{8} = \frac{35 \times 2}{8 \times 2} = \frac{70}{16} \]
Step 3: Calculate the total length of wire used
Now we add the lengths used for both jobs: \[ \frac{41}{16} + \frac{70}{16} = \frac{41 + 70}{16} = \frac{111}{16} \]
Step 4: Subtract from the total length of wire
Now we subtract the total amount of wire used from the initial total: \[ \frac{168}{16} - \frac{111}{16} = \frac{168 - 111}{16} = \frac{57}{16} \]
Step 5: Convert back to a mixed number (if necessary)
Now, we convert \(\frac{57}{16}\) back to a mixed number: \[ 57 \div 16 = 3 \quad \text{(remainder 9)} \] Thus, \[ \frac{57}{16} = 3 \frac{9}{16} \]
Final Answer: The electrician has \(\boxed{3 \frac{9}{16}}\) inches of wire left.
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