1. The problem is to multiply the mixed number $2 \frac{1}{4}$ by 6. 2. First, convert the mixed number to an improper fraction. Recall that $2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}$. 3. Now multiply the improper fraction by 6: $$\frac{9}{4} \times 6 = \frac{9}{4} \times \frac{6}{1}$$. 4. Multiply the numerators and denominators: $$\frac{9 \times 6}{4 \times 1} = \frac{54}{4}$$. 5. Simplify the fraction by dividing numerator and denominator by 2: $$\frac{54 \div 2}{4 \div 2} = \frac{27}{2}$$. 6. Convert the improper fraction back to a mixed number: $27 \div 2 = 13$ remainder $1$, so the answer is $13 \frac{1}{2}$. Final answer: $13 \frac{1}{2}$.