1. Problem: Simplify the product $\frac{7}{33}\times\frac{27}{14}$.\n 2. Formula and rules: Use the rule $\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$.\n Multiply numerators and multiply denominators, and simplify by canceling common factors before multiplying to keep numbers small.\n 3. Original expression:\n $$\frac{7}{33}\times\frac{27}{14}$$\n 4. Cross-cancel common factors between numerators and denominators to simplify before multiplying.\n First cancel 7 and 14 (factor 7):\n $$\frac{\cancel{7}}{33}\times\frac{27}{\cancel{14}}$$\n After canceling 7 with 14 (dividing both by 7) we get\n $$=\frac{1}{33}\times\frac{27}{2}$$\n 5. Now cross-cancel 27 and 33 (common factor 3):\n $$=\frac{1}{\cancel{33}}\times\frac{\cancel{27}}{2}$$\n Dividing both by 3 gives\n $$=\frac{1}{11}\times\frac{9}{2}$$\n 6. Multiply the simplified numerators and denominators.\n $$=\frac{1\times9}{11\times2}=\frac{9}{22}$$\n 7. Final answer: The completely simplified product is $\frac{9}{22}$.\n