Express the following as a common fraction in simplest form: $$\frac{9 \cdot 3 + 8}{4 \cdot 3 + 8}.$$
First, I'll simplify the numerator by performing the multiplication and addition: $$\frac{9 \cdot 3 + 8}{4 \cdot 3 + 8} = \frac{27 + 8}{12 + 8} = \frac{35}{20}.$$ I can list the factors of 35 as 1, 5, 7, and 35, and the factors of 20 as 1, 2, 4, 5, 10, and 20. I can see that 5 is a common factor of both 35 and 20, so I can divide both the numerator and denominator by 5 to reduce the fraction: $$\frac{35}{20} = \frac{35 \div 5}{20 \div 5} = \frac{7}{4}.$$ Now, I can see that 7 and 4 have no common factors other than 1, so the fraction is in its simplest form.

