Half the pizza is $$\frac{1}{2} \times 36 = 18$$ slices. To find which calculation gives a result closest to 18, we need the fraction closest to $$\frac{1}{2}$$. Converting to compare with $$\frac{1}{2} = \frac{6}{12}$$: $$\frac{5}{12}$$ differs by $$\frac{1}{12}$$, $$\frac{1}{3} = \frac{4}{12}$$ differs by $$\frac{2}{12}$$, and $$\frac{3}{8} = \frac{9}{24} = \frac{4.5}{12}$$ differs by $$\frac{1.5}{12}$$. Anna's $$\frac{5}{12}$$ is closest to $$\frac{1}{2}$$. Choice A uses incorrect reasoning about numerators. Choice B focuses on simplicity rather than proximity to $$\frac{1}{2}$$. Choice C incorrectly identifies which fraction is closest.

When you see a problem about price reductions using multiplication, you need to understand what those fractions represent. Both stores are multiplying the original price by a fraction less than 1, which creates a sale price. The key insight is that the smaller the fraction, the bigger the discount and the lower the final price.

To compare $$\frac{4}{5}$$ and $$\frac{7}{9}$$ without calculating, convert them to decimals or find a common denominator. Using decimals: $$\frac{4}{5} = 0.8$$ and $$\frac{7}{9} ≈ 0.778$$. Since $$\frac{7}{9}$$ is smaller, multiplying by it gives a lower sale price. You can also think about distance from 1: $$\frac{4}{5}$$ is $$\frac{1}{5}$$ away from 1, while $$\frac{7}{9}$$ is $$\frac{2}{9}$$ away from 1. Since $$\frac{2}{9} > \frac{1}{5}$$, the fraction $$\frac{7}{9}$$ is farther from 1.

Choice A is correct because $$\frac{7}{9}$$ is indeed farther from 1 than $$\frac{4}{5}$$, creating the lower price. Choice B incorrectly focuses on denominator size, which doesn't determine the fraction's value. Choice C has the logic backward—it correctly identifies that being farther from 1 creates a lower price, but wrongly claims $$\frac{4}{5}$$ is farther from 1. Choice D misses that different fractions less than 1 create different sale prices.

Remember: when comparing discounts given as multipliers, the smaller the fraction, the bigger the discount. Focus on which fraction is closer to zero, not which has the smaller denominator.

Emma is incorrect. When multiplying by the same whole number (42), the fraction closer to 1 produces the larger product. $$\frac{11}{12}$$ is closer to 1 than $$\frac{5}{6}$$ (since $$\frac{11}{12} = \frac{22}{24}$$ and $$\frac{5}{6} = \frac{20}{24}$$), so $$\frac{11}{12} \times 42$$ will be larger. Choice A incorrectly generalizes about denominators. Choice B focuses on computational ease rather than product size. Choice D incorrectly suggests fractions with different denominators cannot be compared.

When comparing fractions without calculating exact amounts, you need to compare each fraction to the benchmark you're measuring against. Here, the recipe needs $$\frac{3}{4}$$ of a bag, so you're comparing $$\frac{7}{8}$$ and $$\frac{2}{3}$$ to $$\frac{3}{4}$$.

To compare fractions quickly, look for common denominators or convert to decimals mentally. For $$\frac{7}{8}$$ versus $$\frac{3}{4}$$: since $$\frac{3}{4} = \frac{6}{8}$$, and $$\frac{7}{8} > \frac{6}{8}$$, Lisa's $$\frac{7}{8}$$ bag has more flour than the recipe needs.

For $$\frac{2}{3}$$ versus $$\frac{3}{4}$$: convert to twelfths. $$\frac{2}{3} = \frac{8}{12}$$ and $$\frac{3}{4} = \frac{9}{12}$$. Since $$\frac{8}{12} < \frac{9}{12}$$, the $$\frac{2}{3}$$ bag has less flour than needed.

Choice A is wrong because it ignores the comparison to $$\frac{3}{4}$$ and incorrectly assumes that fractions less than 1 are automatically insufficient. Choice B incorrectly calls both fractions "large" without comparing them to the specific requirement of $$\frac{3}{4}$$. Choice C reverses which bag has more and which has less than the recipe needs.

The correct answer is D: the $$\frac{7}{8}$$ bag has more than needed, while the $$\frac{2}{3}$$ bag has less.

Remember: when comparing fractions, always compare to the specific benchmark given in the problem, not just to 1 or to general ideas about "large" or "small" fractions.

Since all products use the same factor (24), we compare the other factors. $$\frac{1}{8} < \frac{7}{8} < 1\frac{1}{8}$$, so the products follow the same order. $$\frac{1}{8} \times 24$$ is smallest because $$\frac{1}{8}$$ is much less than 1. $$\frac{7}{8} \times 24$$ is next because $$\frac{7}{8} < 1$$. $$1\frac{1}{8} \times 24$$ is largest because $$1\frac{1}{8} > 1$$. The other choices incorrectly order the factors or misunderstand how factor size affects product size.

David's conclusion is correct but his reasoning is flawed. $$1\frac{2}{7} = \frac{9}{7}$$, and since $$\frac{9}{7} > \frac{8}{7}$$, the first product will indeed be larger. However, this is because $$\frac{9}{7}$$ represents a larger value than $$\frac{8}{7}$$, not because of the mixed number vs. improper fraction format. The format doesn't determine size—the actual value does. Choice A and B accept the flawed reasoning. Choice D incorrectly reverses the relationship between improper fractions and mixed numbers.

When multiplying a whole number by a fraction, the closer the fraction is to 1, the closer the product will be to the original whole number. Since $$\frac{5}{6} = 0.833...$$ is closer to 1 than $$\frac{3}{4} = 0.75$$, the product $$\frac{5}{6} \times 8$$ will be closer to 8. Choice A incorrectly identifies which fraction is closer to 1. Choice C ignores the effect of different fraction multipliers. Choice D confuses smaller fractions with closer-to-8 products.

Both students are correct about what they claimed. Tyler correctly identifies that $$\frac{6}{5} \times 30 = 30 \times \frac{6}{5}$$ by the commutative property of multiplication—the products are equal to each other. Sarah correctly identifies that both products are greater than 30 because $$\frac{6}{5} > 1$$, so multiplying 30 by $$\frac{6}{5}$$ gives a result larger than 30. Choice A incorrectly suggests we cannot determine the relationship to 30. Choice B incorrectly doubts the equality of the products. Choice D incorrectly claims the products equal 30.

Carlos is correct. When multiplying a whole number by a fraction less than 1, the product is always smaller than the whole number. Both $$\frac{2}{3} < 1$$ and $$\frac{4}{5} < 1$$, so both products will be less than 15 and 12 respectively. Choice A overgeneralizes incorrectly about all fractions. Choice C incorrectly thinks being close to 1 means the product exceeds the whole number. Choice D incorrectly states that multiplication always increases values.

For a product between 18 and 20, we need a fraction between $$\frac{9}{10}$$ and 1. $$\frac{19}{20} = 0.95$$ is between $$\frac{9}{10} = 0.9$$ and 1, so $$\frac{19}{20} \times 20$$ will be between 18 and 20. Choice A gives $$\frac{8}{10} \times 20 = 16$$, which is less than 18. Choice C gives a product greater than 20. Choice D also gives a product greater than 20 since $$\frac{11}{10} > 1$$.