Question: Return the original user prompt in the "prompt" field. Wrap every mathematical expression, number used in a calculation, equation, fraction, variable, or formula in $...$ for inline math or $$...$$ for display math.
1. **State the problem:** Simplify the expression $$\frac{170+50x}{x} \div \frac{120+30x}{x}$$.
2. **Recall the division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal.
3. **Rewrite the division as multiplication:**
$$\frac{170+50x}{x} \times \frac{x}{120+30x}$$
4. **Simplify the expression:**
$$\frac{170+50x}{x} \times \frac{x}{120+30x} = \frac{(170+50x) \cancel{x}}{\cancel{x} (120+30x)}$$
5. **Cancel the common factor $x$ in numerator and denominator:**
$$= \frac{170+50x}{120+30x}$$
6. **Factor numerator and denominator:**
$$170+50x = 10(17+5x)$$
$$120+30x = 30(4+x)$$
7. **Rewrite the fraction with factored terms:**
$$\frac{10(17+5x)}{30(4+x)}$$
8. **Simplify the fraction by dividing numerator and denominator by 10:**
$$\frac{\cancel{10}(17+5x)}{3 \times \cancel{10} (4+x)} = \frac{17+5x}{3(4+x)}$$
**Final answer:**
$$\boxed{\frac{17+5x}{3(4+x)}}$$