1. **State the problem:** We have 25 gems with three different cuts and three different colors: amethysts, emeralds, and rubies. We want to find the probability that a randomly selected gem is either an emerald or a ruby. 2. **Formula for probability:** $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$ 3. **Important rule:** Since the gems are categorized by color, and we want emeralds or rubies, the favorable outcomes are all emeralds plus all rubies. 4. **Calculate the number of favorable outcomes:** Since the problem does not specify the distribution of colors, we assume the gems are evenly distributed among the three colors. Number of gems per color = $\frac{25}{3} \approx 8.33$ (not an integer, but we proceed with the fraction for probability calculation). 5. **Calculate the probability:** Number of emeralds or rubies = $8.33 + 8.33 = 16.66$ Total gems = 25 $$\text{Probability} = \frac{16.66}{25} = 0.6664$$ 6. **Interpretation:** The probability that a randomly selected gem is an emerald or a ruby is approximately $0.6664$ or $\frac{2}{3}$ if the gems are evenly distributed. **Final answer:** $$\boxed{\frac{2}{3}}$$