To find how many miles farther it is from the school to the library than from the library to Town Hall, we can subtract the distance from the library to Town Hall from the distance from the school to the library.
- The distance from the school to the library is \(3 \frac{1}{2}\) miles.
- The distance from the library to Town Hall is \(1 \frac{1}{4}\) miles.
Now, we will convert these mixed numbers into improper fractions for easier subtraction:
- \(3 \frac{1}{2} = \frac{7}{2}\) (since \(3 \times 2 + 1 = 7\))
- \(1 \frac{1}{4} = \frac{5}{4}\) (since \(1 \times 4 + 1 = 5\))
Next, we need to find a common denominator to subtract the fractions. The least common denominator of 2 and 4 is 4.
Now, we convert \(\frac{7}{2}\) to have a denominator of 4:
\[ \frac{7}{2} = \frac{7 \times 2}{2 \times 2} = \frac{14}{4} \]
Now we can perform the subtraction:
\[ \frac{14}{4} - \frac{5}{4} = \frac{14 - 5}{4} = \frac{9}{4} \]
Now we convert \(\frac{9}{4}\) back to a mixed number:
\(\frac{9}{4} = 2 \frac{1}{4}\) because \(9\) divided by \(4\) is \(2\) with a remainder of \(1\).
Thus, the distance from the school to the library is \(2 \frac{1}{4}\) miles farther than from the library to Town Hall.
Final Answer: \(2 \frac{1}{4}\) miles.