1. The problem is to calculate the product of two fractions: $\frac{5}{8} \times \frac{2}{3}$ and simplify the result. 2. The formula for multiplying fractions is: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ where $a$, $b$, $c$, and $d$ are integers and $b, d \neq 0$. 3. Applying the formula: $$\frac{5}{8} \times \frac{2}{3} = \frac{5 \times 2}{8 \times 3} = \frac{10}{24}$$ 4. Now simplify $\frac{10}{24}$ by dividing numerator and denominator by their greatest common divisor (GCD). The GCD of 10 and 24 is 2. 5. Divide numerator and denominator by 2: $$\frac{10 \div 2}{24 \div 2} = \frac{5}{12}$$ 6. The fraction $\frac{5}{12}$ is in simplest form because 5 and 12 have no common divisors other than 1. **Final answer:** $$\frac{5}{12}$$