$5 \frac{1}{4}-2 \frac{5}{16}$

Step 1 ：Convert the mixed numbers to improper fractions: $5 \frac{1}{4} = \frac{21}{4}$ and $2 \frac{5}{16} = \frac{37}{16}$

Step 2 ：Find a common denominator: $4$ and $16$ have a common denominator of $64$

Step 3 ：Convert the fractions to have the common denominator: $\frac{21}{4} = \frac{21 \times 16}{4 \times 16} = \frac{336}{64}$ and $\frac{37}{16} = \frac{37 \times 4}{16 \times 4} = \frac{148}{64}$

Step 4 ：Subtract the numerators: $336 - 148 = 188$

Step 5 ：Write the result as a fraction with the common denominator: $\frac{188}{64}$

Step 6 ：Simplify the fraction: $\frac{188}{64} = \frac{47}{16}$

Step 7 ：Convert the improper fraction back to a mixed number: $\frac{47}{16} = 2 \frac{15}{16}$

Step 8 ：\(\boxed{2 \frac{15}{16}}\)