1. The problem is to subtract the mixed numbers $13 \frac{1}{4}$ and $3 \frac{3}{4}$.\n\n2. First, convert the mixed numbers to improper fractions.\n\n$$13 \frac{1}{4} = \frac{13 \times 4 + 1}{4} = \frac{52 + 1}{4} = \frac{53}{4}$$\n$$3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$$\n\n3. Now subtract the improper fractions: $$\frac{53}{4} - \frac{15}{4} = \frac{53 - 15}{4} = \frac{38}{4}$$\n\n4. Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2: $$\frac{\cancel{38}^{19}}{\cancel{4}^{2}} = \frac{19}{2}$$\n\n5. Convert the improper fraction back to a mixed number: $$\frac{19}{2} = 9 \frac{1}{2}$$\n\n6. Therefore, the answer is $9 \frac{1}{2}$.