Simplify (2 1/4 - 1 5/8) ÷ 2 1/6

Step 1: Convert all mixed numbers to improper fractions. $2\frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4}$ $1\frac{5}{8} = \frac{(1 \times 8) + 5}{8} = \frac{13}{8}$ $2\frac{1}{6} = \frac{(2 \times 6) + 1}{6} = \frac{13}{6}$

Step 2: Perform the subtraction inside the parentheses. To subtract $\frac{9}{4} - \frac{13}{8}$, find a common denominator, which is 8. $\frac{9}{4} - \frac{13}{8} = \frac{9 \times 2}{4 \times 2} - \frac{13}{8} = \frac{18}{8} - \frac{13}{8} = \frac{18 - 13}{8} = \frac{5}{8}$

Step 3: Perform the division. Now, divide the result from Step 2 by the third fraction. $\frac{5}{8} \div \frac{13}{6}$ To divide by a fraction, multiply by its reciprocal. $\frac{5}{8} \times \frac{6}{13}$

Step 4: Multiply the fractions and simplify. $\frac{5 \times 6}{8 \times 13} = \frac{30}{104}$ Both the numerator and the denominator are divisible by 2. $\frac{30 \div 2}{104 \div 2} = \frac{15}{52}$

The simplified expression is $\boxed{\frac{15}{52}}$.