1. The problem is to solve the proportion $$\frac{x}{78} = \frac{8}{13}$$ and find the value of $x$. 2. The formula for solving proportions is to cross multiply: $$a/b = c/d \implies ad = bc$$. 3. Applying cross multiplication here: $$x \times 13 = 8 \times 78$$. 4. Calculate the right side: $$8 \times 78 = 624$$, so we have $$13x = 624$$. 5. To isolate $x$, divide both sides by 13: $$x = \frac{624}{13}$$. 6. Show the cancellation step: $$x = \frac{\cancel{624}}{\cancel{13}}$$ (simplify by dividing 624 by 13). 7. Perform the division: $$x = 48$$. 8. Therefore, the missing number $x$ is 48.