To find out how much longer the track will be if it is extended from \( \frac{1}{5} \) of a mile to \( \frac{1}{4} \) of a mile, you can subtract the current length from the new length.
The lengths in fractions are:
- Current length: \( \frac{1}{5} \) mile
- New length: \( \frac{1}{4} \) mile
To subtract these fractions, first find a common denominator. The least common multiple of 5 and 4 is 20.
Now, convert both fractions to have the common denominator of 20:
\[ \frac{1}{5} = \frac{4}{20} \] \[ \frac{1}{4} = \frac{5}{20} \]
Now subtract the two fractions:
\[ \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \]
Therefore, the track will be extended by \( \frac{1}{20} \) of a mile.
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