1. **State the problem:** We want to find the difference between $1 \frac{1}{4}$ and $\frac{1}{5}$. 2. **Convert mixed number to improper fraction:** $$1 \frac{1}{4} = \frac{4 \times 1 + 1}{4} = \frac{5}{4}$$ 3. **Subtract fractions:** To subtract $\frac{5}{4} - \frac{1}{5}$, find a common denominator. The least common denominator (LCD) of 4 and 5 is 20. 4. **Convert each fraction to have denominator 20:** $$\frac{5}{4} = \frac{5 \times 5}{4 \times 5} = \frac{25}{20}$$ $$\frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20}$$ 5. **Perform the subtraction:** $$\frac{25}{20} - \frac{4}{20} = \frac{25 - 4}{20} = \frac{21}{20}$$ 6. **Simplify the result if possible:** $\frac{21}{20}$ is an improper fraction and can be written as the mixed number: $$1 \frac{1}{20}$$ **Final answer:** $$1 \frac{1}{20}$$