Simplify this 3 2/3 × 7 1/2 ÷ 1 2/9

$$3\frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$$ $$7\frac{1}{2} = \frac{(7 \times 2) + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2}$$ $$1\frac{2}{9} = \frac{(1 \times 9) + 2}{9} = \frac{9 + 2}{9} = \frac{11}{9}$$

The expression becomes:

$$\frac{11}{3} \times \frac{15}{2} \div \frac{11}{9}$$

$$\frac{11}{3} \times \frac{15}{2} = \frac{11 \times 15}{3 \times 2} = \frac{165}{6}$$

Simplify the fraction $\frac{165}{6}$ by dividing the numerator and denominator by their greatest common divisor, which is 3:

$$\frac{165 \div 3}{6 \div 3} = \frac{55}{2}$$

To divide by a fraction, multiply by its reciprocal.

$$\frac{55}{2} \div \frac{11}{9} = \frac{55}{2} \times \frac{9}{11}$$

$$\frac{55}{2} \times \frac{9}{11} = \frac{55 \times 9}{2 \times 11}$$

We can simplify by canceling out the common factor of 11 (since $55 = 5 \times 11$):

$$\frac{(5 \times 11) \times 9}{2 \times 11} = \frac{5 \times 9}{2} = \frac{45}{2}$$

$$\frac{45}{2} = 22\frac{1}{2}$$

The simplified expression is:

$$22\frac{1}{2}$$