To find out how many miles farther it is from the school to the library than from the library to Town Hall, we need to 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.
First, let's convert these mixed numbers to improper fractions:
- \(3 \frac{1}{2} = \frac{7}{2}\) (because \(3 \times 2 + 1 = 7\))
- \(1 \frac{1}{4} = \frac{5}{4}\) (because \(1 \times 4 + 1 = 5\))
Now we subtract the distance from the library to Town Hall from the distance from the school to the library:
\[ \frac{7}{2} - \frac{5}{4} \]
To subtract these fractions, we need a common denominator. The least common multiple of 2 and 4 is 4.
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 convert \(\frac{9}{4}\) back to a mixed number:
\(\frac{9}{4} = 2 \frac{1}{4}\)
Thus, it is \(2 \frac{1}{4}\) miles farther from the school to the library than from the library to Town Hall.