1. **Problem:** Find the difference $1 \frac{2}{5} - \frac{3}{4}$.\n\n2. **Formula to use:** To subtract fractions, first write any mixed number as an improper fraction, then find a common denominator: $$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$$\n\n3. **Convert the mixed number:** $$1 \frac{2}{5} = \frac{7}{5}$$\n\n4. **Find a common denominator:** The denominators are $5$ and $4$, so the common denominator is $20$.\n\n5. **Rewrite each fraction:** $$\frac{7}{5} = \frac{28}{20}$$ $$\frac{3}{4} = \frac{15}{20}$$\n\n6. **Subtract the fractions:** $$\frac{28}{20} - \frac{15}{20} = \frac{13}{20}$$\n\n7. **Final answer:** $$\frac{13}{20}$$