Dr. Moreno, a renowned geologist, is studying six newly discovered minerals labeled A through F. Each mineral has a unique hardness, and Dr. Moreno wants to rank them from hardest (1st) to softest (6th). After conducting a series of scratch tests, she records the following observations: A is harder than C and D. B is softer than only one other mineral. E is softer than C but harder than F. D is softer than A but harder than C. F is the softest. Based on the above information, select for 1st the mineral ranked the hardest, and for 3rd the mineral ranked 3rd.

Solution: Based on the information 1. A is the hardest. 2. B is softer than A but harder than D. 3. D is softer than B and A but harder than C. 4. C is softer than A, B, and D but harder than E. 5. E is softer than A, B, D, and C but harder than F. 6. F is the softest one. Hence, the first mineral ranked the hardest is A, and the third mineral ranked the third hardest is D.

1. The question asks us to rank the minerals by hardness and figure out the 1st and 3rd in placement . 2. Dr. Moreno made some observations that can be made transformed into inequalities to make it easier: - A is harder than C and D. A > C,D - E is softer than C but harder than F. C > E > F - D is softer than A but harder than C. A > D > C 3. Combining these inequalities together gives us A > D > C > E > F. 4. The only mineral missing is B, for which it is known that “B is softer than only one other mineral.” So, A > B > D > C > E > F. 5. “F is the softest” confirms that our ranking is correct. 6. Our answer will be: 1st - A and 3rd - D .