1. **State the problem:** Solve the equation $$\frac{140+35x}{150+30x} = \frac{28}{27}$$ for $x$. 2. **Use the cross-multiplication rule:** When two fractions are equal, their cross products are equal. So, $$ (140+35x) \times 27 = (150+30x) \times 28 $$ 3. **Expand both sides:** $$ 27 \times 140 + 27 \times 35x = 28 \times 150 + 28 \times 30x $$ $$ 3780 + 945x = 4200 + 840x $$ 4. **Group like terms:** Move all terms involving $x$ to one side and constants to the other. $$ 945x - 840x = 4200 - 3780 $$ $$ 105x = 420 $$ 5. **Solve for $x$:** $$ x = \frac{420}{105} = 4 $$ 6. **Final answer:** $$ \boxed{4} $$ This means the value of $x$ that satisfies the equation is 4.