To find out how much of the pencil has been used, you can subtract the remaining length of the pencil from the original length.
The original length of the pencil is \(6 \frac{1}{2}\) inches, which can be converted to an improper fraction: \[ 6 \frac{1}{2} = \frac{13}{2} \text{ inches} \] The remaining length of the pencil is \(5 \frac{1}{4}\) inches, which can also be converted to an improper fraction: \[ 5 \frac{1}{4} = \frac{21}{4} \text{ inches} \]
Now we need to perform the subtraction: \[ \frac{13}{2} - \frac{21}{4} \]
First, we need a common denominator. The least common multiple of \(2\) and \(4\) is \(4\). Convert \(\frac{13}{2}\) to a fraction with a denominator of \(4\): \[ \frac{13}{2} = \frac{13 \times 2}{2 \times 2} = \frac{26}{4} \]
Now perform the subtraction: \[ \frac{26}{4} - \frac{21}{4} = \frac{26 - 21}{4} = \frac{5}{4} \]
Finally, convert \(\frac{5}{4}\) back to a mixed number: \[ \frac{5}{4} = 1 \frac{1}{4} \]
So, the amount of the pencil that has been used is \(1 \frac{1}{4}\) inches.