1. **State the problem:** Rahul spends $\frac{5}{9}$ of his vacation at summer camp, then $\frac{3}{4}$ of the remaining time at his grandparents' home, and the remaining 7 days at the seashore. We need to find the total number of days of his summer vacation. 2. **Define variables:** Let the total vacation days be $x$. 3. **Calculate time spent at summer camp:** He spends $\frac{5}{9}x$ days at summer camp. 4. **Calculate remaining time after summer camp:** Remaining days = $x - \frac{5}{9}x = \frac{4}{9}x$. 5. **Calculate time spent at grandparents' home:** He spends $\frac{3}{4}$ of the remaining $\frac{4}{9}x$ days there, which is $$\frac{3}{4} \times \frac{4}{9}x = \frac{3}{9}x = \frac{1}{3}x.$$ 6. **Calculate remaining time at seashore:** Remaining days after grandparents' home = $\frac{4}{9}x - \frac{1}{3}x$. 7. **Simplify remaining days:** Convert $\frac{1}{3}x$ to ninths: $\frac{1}{3}x = \frac{3}{9}x$, so $$\frac{4}{9}x - \frac{3}{9}x = \frac{1}{9}x.$$ 8. **Set remaining days equal to 7:** The problem states the remaining days at the seashore is 7, so $$\frac{1}{9}x = 7.$$ 9. **Solve for $x$:** Multiply both sides by 9: $$x = 7 \times 9 = 63.$$ 10. **Conclusion:** Rahul's total summer vacation is 63 days. **Final answer:** $\boxed{63}$ days.