1. **State the problem:** We need to calculate $9 \div 1 \frac{3}{4}$ and express the answer as a mixed number in its lowest terms. 2. **Convert the mixed number to an improper fraction:** $$1 \frac{3}{4} = \frac{4 \times 1 + 3}{4} = \frac{7}{4}$$ 3. **Rewrite the division as multiplication by the reciprocal:** $$9 \div \frac{7}{4} = 9 \times \frac{4}{7}$$ 4. **Multiply the fractions:** $$9 \times \frac{4}{7} = \frac{9 \times 4}{7} = \frac{36}{7}$$ 5. **Convert the improper fraction to a mixed number:** Divide 36 by 7: $$36 \div 7 = 5 \text{ remainder } 1$$ So, $$\frac{36}{7} = 5 \frac{1}{7}$$ **Final answer:** $$5 \frac{1}{7}$$