Simplify: 15/4 x 2 2/3 ÷ 5/6 x 3/10

Step 1: Convert the mixed number to an improper fraction. $2\frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$

Step 2: Rewrite the expression with all fractions. $\frac{15}{4} \times \frac{8}{3} \div \frac{5}{6} \times \frac{3}{10}$

Step 3: Change the division operation to multiplication by the reciprocal of the fraction being divided. $\frac{15}{4} \times \frac{8}{3} \times \frac{6}{5} \times \frac{3}{10}$

Step 4: Multiply the fractions. Cancel common factors before multiplying to simplify the calculation. $\frac{15}{4} \times \frac{8}{3} \times \frac{6}{5} \times \frac{3}{10} = \frac{15 \times 8 \times 6 \times 3}{4 \times 3 \times 5 \times 10}$ Cancel $3$ from numerator and denominator: $= \frac{15 \times 8 \times 6 \times 1}{4 \times 1 \times 5 \times 10}$ Cancel $4$ with $8$ (leaving $2$ in the numerator): $= \frac{15 \times 2 \times 6 \times 1}{1 \times 1 \times 5 \times 10}$ Cancel $5$ with $15$ (leaving $3$ in the numerator): $= \frac{3 \times 2 \times 6 \times 1}{1 \times 1 \times 1 \times 10}$ Multiply the remaining terms: $= \frac{36}{10}$

Step 5: Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is $2$. $= \frac{36 \div 2}{10 \div 2} = \frac{18}{5}$

The final answer is $\boxed{\frac{18}{5}}$.