1. **State the problem:** Jack sold $\frac{3}{7}$ of 63 books, and Amelia sold $\frac{6}{7}$ of 42 books. We need to find the difference between the numbers of books they each sold. 2. **Formula used:** To find the number of books sold, multiply the fraction sold by the total number of books. 3. **Calculate books Jack sold:** $$\text{Jack's books sold} = \frac{3}{7} \times 63$$ Simplify: $$= \frac{3}{\cancel{7}} \times \cancel{63}9 = 3 \times 9 = 27$$ 4. **Calculate books Amelia sold:** $$\text{Amelia's books sold} = \frac{6}{7} \times 42$$ Simplify: $$= \frac{6}{\cancel{7}} \times \cancel{42}6 = 6 \times 6 = 36$$ 5. **Find the difference:** $$\text{Difference} = 36 - 27 = 9$$ **Final answer:** The difference between the numbers of books they have each sold is 9.