1. **Stating the problem:** We need to determine which fraction is greater between $\frac{5}{8}$ and $\frac{2}{3}$.\n\n2. **Formula and rules:** To compare fractions, we can find a common denominator or convert them to decimals. The fraction with the larger value is greater.\n\n3. **Find common denominator:** The denominators are 8 and 3. The least common denominator (LCD) is $24$.\n\n4. **Convert fractions:**\n$\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$\n$\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}$\n\n5. **Compare numerators:** Since $15 < 16$, $\frac{5}{8} < \frac{2}{3}$.\n\n6. **Conclusion:** $\frac{2}{3}$ is greater than $\frac{5}{8}$.\n