Here are the mathematical translations of the given English statements:

1. The difference of nine and three is six
   $9 - 3 = 6$

2. The product of twice a number and six is eight plus a number
   $2x \cdot 6 = 8 + x$
   (Let $x$ represent the unknown number.)

3. Sixty percent of a number is fifteen
   $0.60x = 15$
   (Let $x$ represent the unknown number.)

4. The ratio of a number and eighteen is two-thirds
   $\frac{x}{18} = \frac{2}{3}$
   (Let $x$ represent the unknown number.)

5. Four times the difference of eight and a number is the same as the ratio of five and twelve
   $4(8 - x) = \frac{5}{12}$
   (Let $x$ represent the unknown number.)

Would you like further details on solving any of these? Here are 5 follow-up questions for more practice:

1. How do you solve for $x$ in $2x \cdot 6 = 8 + x$?
2. What is the solution to $0.60x = 15$?
3. How can you solve $\frac{x}{18} = \frac{2}{3}$?
4. How would you simplify $4(8 - x) = \frac{5}{12}$?
5. What are other methods to translate word problems into mathematical equations?

Tip: Always define the unknown variable first when translating word problems into algebraic expressions.