1. **State the problem:** Simplify the expression $\frac{2}{3} \times \frac{3}{5}$.\n\n2. **Recall the multiplication rule for fractions:** To multiply two fractions, multiply the numerators together and the denominators together. The formula is: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.$$\n\n3. **Apply the formula:** Multiply the numerators: $2 \times 3 = 6$. Multiply the denominators: $3 \times 5 = 15$. So, $$\frac{2}{3} \times \frac{3}{5} = \frac{6}{15}.$$\n\n4. **Simplify the fraction:** Find the greatest common divisor (GCD) of 6 and 15, which is 3. Divide numerator and denominator by 3: $$\frac{\cancel{6}^2}{\cancel{15}^5} = \frac{2}{5}.$$\n\n5. **Final answer:** The simplified product is $\frac{2}{5}$.