1. The problem asks to explain why the product in the equation $8 \times 1 \frac{1}{8} = 5 \frac{1}{8}$ is not possible. 2. First, convert the mixed number $1 \frac{1}{8}$ to an improper fraction: $$1 \frac{1}{8} = \frac{9}{8}$$ 3. Multiply $8$ by $\frac{9}{8}$: $$8 \times \frac{9}{8} = \frac{8}{1} \times \frac{9}{8} = \cancel{\frac{8}{1}} \times \cancel{\frac{9}{8}} = 9$$ 4. The product is $9$, which is not equal to $5 \frac{1}{8}$, so the equation is incorrect. 5. Next, explain why the product in $1 \frac{1}{4} \times 10 = 20 \frac{1}{2}$ is not possible. 6. Convert $1 \frac{1}{4}$ to an improper fraction: $$1 \frac{1}{4} = \frac{5}{4}$$ 7. Multiply $\frac{5}{4}$ by $10$: $$\frac{5}{4} \times 10 = \frac{5}{4} \times \frac{10}{1} = \cancel{\frac{5}{4}} \times \cancel{\frac{10}{1}} = \frac{50}{4} = 12 \frac{1}{2}$$ 8. The product is $12 \frac{1}{2}$, which is not equal to $20 \frac{1}{2}$, so the equation is incorrect. 9. For the cook's sauce problem: A cook uses $5 \frac{2}{5}$ cups of sauce per tray. How many cups for 3 trays? 10. Convert $5 \frac{2}{5}$ to an improper fraction: $$5 \frac{2}{5} = \frac{27}{5}$$ 11. Multiply by 3 trays: $$\frac{27}{5} \times 3 = \frac{27}{5} \times \frac{3}{1} = \frac{81}{5} = 16 \frac{1}{5}$$ 12. So, $16 \frac{1}{5}$ cups of sauce are needed for 3 trays. 13. Rounding $567.01$ to the ones place: The digit in the tenths place is 0, which is less than 5, so round down. 14. Rounded number is $567$, which is smaller than $568$. 15. Rounding $222.978$ to the hundredths place: The digit in the thousandths place is 8, which is 5 or more, so round up. 16. Rounded number is $222.98$, which is larger than $222.97$. 17. Rounding $628.54$ to the tens place: The digit in the ones place is 8, which is 5 or more, so round up. 18. Rounded number is $630$, which is larger than $628.6$. Final answers: - $8 \times 1 \frac{1}{8} \neq 5 \frac{1}{8}$ because the product is actually $9$. - $1 \frac{1}{4} \times 10 \neq 20 \frac{1}{2}$ because the product is actually $12 \frac{1}{2}$. - Sauce needed for 3 trays: $16 \frac{1}{5}$ cups. - $567.01$ rounded to ones is $567$, smaller than $568$. - $222.978$ rounded to hundredths is $222.98$, larger than $222.97$. - $628.54$ rounded to tens is $630$, larger than $628.6$.