To find the length of the smaller piece, we can subtract the length of the larger piece from the total length of the original rod.
The original rod is 9 1/8 inches long, and the larger piece is 7 3/4 inches long.
First, we need to align the two fractions for easier subtraction. We can convert both measurements to improper fractions.
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Convert 9 1/8 to an improper fraction: \[ 9 \frac{1}{8} = \frac{9 \times 8 + 1}{8} = \frac{72 + 1}{8} = \frac{73}{8} \]
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Convert 7 3/4 to an improper fraction: \[ 7 \frac{3}{4} = \frac{7 \times 4 + 3}{4} = \frac{28 + 3}{4} = \frac{31}{4} \]
Next, we need to make the denominators the same in order to subtract these fractions. The least common denominator (LCD) of 8 and 4 is 8. Now we convert \(\frac{31}{4}\) to have a denominator of 8:
\[ \frac{31}{4} = \frac{31 \times 2}{4 \times 2} = \frac{62}{8} \]
Now we can perform the subtraction:
\[ \text{Length of the smaller piece} = \frac{73}{8} - \frac{62}{8} = \frac{73 - 62}{8} = \frac{11}{8} \]
Finally, we convert \(\frac{11}{8}\) back to a mixed number:
\[ \frac{11}{8} = 1 \frac{3}{8} \]
Thus, the length of the resulting smaller piece is \(1 \frac{3}{8}\) inches.