1. Stating the problem: Calculate the value of $13\ \frac{2}{5} - \frac{3}{5}$. 2. Convert the mixed number to an improper fraction: $13\ \frac{2}{5} = \frac{13 \times 5 + 2}{5} = \frac{65 + 2}{5} = \frac{67}{5}$. 3. Subtract the fractions: $\frac{67}{5} - \frac{3}{5} = \frac{67 - 3}{5} = \frac{64}{5}$. 4. Convert the improper fraction back to a mixed number: $\frac{64}{5} = 12\ \frac{4}{5}$ because $64 \div 5 = 12$ remainder $4$. 5. Final answer: $12\ \frac{4}{5}$. This problem involves subtracting fractions with the same denominator and converting between mixed numbers and improper fractions, which is a key skill in fraction arithmetic.