1. **State the problem:** We need to use three digits from the cards {6, 0, 5, 9, 8} to create a decimal number $0.\text{abc}$ such that it satisfies $$\frac{57}{100} < 0.\text{abc} < \frac{64}{100}$$ which means $$0.57 < 0.\text{abc} < 0.64$$ 2. **Analyze constraints:** The decimal must be greater than 0.57 and less than 0.64. 3. **Step-by-step search:** - The first digit after decimal (a) must be 5 or 6 because digits 0-9 are available. - If we pick $a=5$, then the decimal looks like 0.5bc and must be greater than 0.57, so digits b and c combined must make it > 0.57. - Using available digits: 6,0,5,9,8 Try 0.58x: 0.58 is 0.58>0.57 and less than 0.64, so that's good. Use 8 from the digits, and then for c, pick any remaining digit (e.g., 0 or 5). Trying 0.589 (digits 5,8,9): 0.589 is greater than 0.57 and less than 0.64. 4. **Answer:** A valid decimal is 0.589 using digits 5, 8, and 9 from the cards. 5. **Summary:** - Three digits chosen: 5, 8, 9 - Decimal formed: $0.589$ - Inequality check: $0.57 < 0.589 < 0.64$ (True) Hence, $\boxed{0.589}$ is a valid decimal number meeting the criteria.