To solve the problem, we need to subtract the length of rope that Liam cut off from the original length of the rope.
Liam's original rope length: \( 7 \frac{13}{16} \) feet
Length cut off: \( 2 \frac{2}{3} \) feet
First, we need to convert both mixed numbers to improper fractions so we can perform the subtraction easily.
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Convert \( 7 \frac{13}{16} \) to an improper fraction: \[ 7 \frac{13}{16} = \frac{(7 \times 16) + 13}{16} = \frac{112 + 13}{16} = \frac{125}{16} \]
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Convert \( 2 \frac{2}{3} \) to an improper fraction: \[ 2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \]
Next, we need a common denominator to subtract \( \frac{8}{3} \) from \( \frac{125}{16} \). The least common multiple of 16 and 3 is 48.
Now we convert each fraction:
\[ \frac{125}{16} \text{ to the common denominator of } 48: \frac{125 \times 3}{16 \times 3} = \frac{375}{48} \]
\[ \frac{8}{3} \text{ to the common denominator of } 48: \frac{8 \times 16}{3 \times 16} = \frac{128}{48} \]
Now we can subtract the two fractions:
\[ \frac{375}{48} - \frac{128}{48} = \frac{375 - 128}{48} = \frac{247}{48} \]
Now we want to convert \( \frac{247}{48} \) back to a mixed number:
- Divide 247 by 48, which gives us 5 as the whole number (since \( 48 \times 5 = 240 \)).
- Subtract \( 240 \) from \( 247 \) to get the remainder, which is \( 7 \).
So, \( \frac{247}{48} = 5 \frac{7}{48} \).
Since the options provided were in feet, and they don't feature \( \frac{7}{48} \), the closest option might be the simple foot equivalent we can see:
- The expected answer based on those options is \( 5 \frac{13}{16} \) would be incorrect based on calculations, but if we simplify it based on choices:
The most suitable answer from the available choices within logical rounding would be 5 feet.
Thus, the correct response is 5 feet.