1. The problem asks us to find which two recurring decimals among $0.3\dot{5}$, $0.35\dot{3}$, $0.35\dot{3}$, and $0.3\dot{5}$ are equivalent. 2. Let's write each recurring decimal as a fraction to compare them. 3. For $0.3\dot{5}$, the decimal is $0.355555\ldots$ where only the 5 repeats. 4. Let $x = 0.355555\ldots$. 5. Multiply by 10 to shift one decimal place: $10x = 3.55555\ldots$. 6. Multiply by 100 to shift two decimal places: $100x = 35.55555\ldots$. 7. Subtract the two equations: $100x - 10x = 35.55555\ldots - 3.55555\ldots$ which gives $90x = 32$. 8. Solve for $x$: $x = \frac{32}{90} = \frac{16}{45}$. 9. For $0.35\dot{3}$, the decimal is $0.353333\ldots$ where only the 3 repeats. 10. Let $y = 0.353333\ldots$. 11. Multiply by 10: $10y = 3.53333\ldots$. 12. Multiply by 100: $100y = 35.3333\ldots$. 13. Subtract: $100y - 10y = 35.3333\ldots - 3.53333\ldots$ which gives $90y = 31.8$. 14. Convert $31.8$ to fraction: $31.8 = \frac{318}{10}$. 15. So, $90y = \frac{318}{10}$, thus $y = \frac{318}{900} = \frac{53}{150}$. 16. We see $0.3\dot{5} = \frac{16}{45}$ and $0.35\dot{3} = \frac{53}{150}$. 17. The decimals $0.3\dot{5}$ and $0.3\dot{5}$ are the same, and $0.35\dot{3}$ and $0.35\dot{3}$ are the same. 18. Therefore, the two equivalent recurring decimals are $0.3\dot{5}$ and $0.3\dot{5}$, and also $0.35\dot{3}$ and $0.35\dot{3}$. Final answer: The pairs $0.3\dot{5}$ and $0.3\dot{5}$, and $0.35\dot{3}$ and $0.35\dot{3}$ are equivalent recurring decimals.