1. The problem is to find the value of $x$ using proportions given as $\frac{40}{800} = \frac{30}{x}$. 2. The formula for solving proportions is to cross-multiply: $a/b = c/d$ implies $a \times d = b \times c$. 3. Applying cross multiplication here: $$40 \times x = 800 \times 30$$ 4. Simplify the right side: $$40x = 24000$$ 5. To isolate $x$, divide both sides by 40: $$\frac{\cancel{40}x}{\cancel{40}} = \frac{24000}{40}$$ 6. Simplify the division: $$x = 600$$ 7. So, the value of $x$ is 600. This means that if 40 corresponds to 800, then 30 corresponds to 600 in the same proportion.