1. **State the problem:** Simplify the division of mixed numbers $1 \frac{1}{4} \div 2 \frac{1}{7}$. 2. **Convert mixed numbers to improper fractions:** $1 \frac{1}{4} = \frac{5}{4}$ $2 \frac{1}{7} = \frac{15}{7}$ 3. **Rewrite the division as multiplication by the reciprocal:** $$\frac{5}{4} \div \frac{15}{7} = \frac{5}{4} \times \frac{7}{15}$$ 4. **Multiply the numerators and denominators:** $$\frac{5 \times 7}{4 \times 15} = \frac{35}{60}$$ 5. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD):** GCD of 35 and 60 is 5, so $$\frac{\cancel{35}^{7}}{\cancel{60}^{12}} = \frac{7}{12}$$ 6. **Final answer:** $$1 \frac{1}{4} \div 2 \frac{1}{7} = \frac{7}{12}$$