Convert all to decimals: $$\frac{5}{8} = 0.625$$, $$62% = 0.62$$, and $$0.627$$ stays as is. Ordering from least to greatest: $$0.62 < 0.625 < 0.627$$, which corresponds to $$62% < \frac{5}{8} < 0.627$$. Choice A incorrectly assumes the fraction is smallest. Choice C reverses the order. Choice D incorrectly places the decimal between the fraction and percent.

When comparing fractions and decimals, you need to convert one form to the other and then make an accurate comparison. This question tests whether you can identify valid mathematical reasoning.

Let's verify the student's work step by step. To convert $$\frac{3}{7}$$ to decimal form, you divide 3 by 7. Performing this division gives you 0.428571428571..., where the digits 428571 repeat indefinitely. The student correctly identified this decimal representation. Now, comparing 0.428571... to 0.42: since 0.428571... is indeed greater than 0.42 (the additional digits 8571... make it larger), the inequality $$\frac{3}{7} > 0.42$$ is true. The student's reasoning is mathematically sound.

Looking at the wrong answers: Answer A suggests the reasoning fails because 0.428571... < 0.43, but this is irrelevant—the comparison was with 0.42, not 0.43. Answer B claims the decimal conversion is wrong, stating it should be $$0.42\overline{857142}$$, but this is actually the same value as 0.428571... (just different notation for the repeating decimal). Answer D incorrectly suggests that fractions and decimals cannot be compared directly, when in fact this is a standard mathematical operation once you convert to the same form.

The correct answer is C because the student performed both the conversion and comparison correctly.

Study tip: When checking decimal conversions of fractions, remember that repeating decimals are exact values, not approximations. Always verify a few decimal places beyond what you're comparing to ensure accuracy.

Convert $$x = 0.4\overline{5}$$ to a fraction: Let $$x = 0.4555...$$ Then $$10x = 4.555...$$ and $$100x = 45.555...$$ So $$100x - 10x = 45.555... - 4.555... = 41$$, giving $$90x = 41$$, so $$x = \frac{41}{90}$$. Therefore $$x = y$$. Choices A and D make false generalizations about the relationship between decimals and fractions. Choice C incorrectly assumes $$0.4\overline{5} < 0.456$$, when actually $$0.4\overline{5} = 0.4555... > 0.456$$.

Convert all to the same form to compare: $$\frac{7}{12} = 0.58\overline{3} = 0.583\overline{3}$$. To convert to percent: $$\frac{7}{12} \times 100% = \frac{700}{12}% = 58\frac{4}{12}% = 58\frac{1}{3}%$$. All three expressions represent the same value. Students might incorrectly think the repeating decimal is different from the fraction, or miscalculate the percentage conversion.

First, convert $$\frac{3}{8}$$ to decimal: $$\frac{3}{8} = 0.375$$. Maria has $$0.4$$ cups, so she has $$0.4 - 0.375 = 0.025$$ cups extra. Converting $$0.025$$ to a fraction: $$0.025 = \frac{25}{1000} = \frac{1}{40}$$. Choice A is wrong because $$0.4 \neq 0.375$$. Choice C gives the correct numerical difference but in decimal form, not fraction form as stated in choice B. Choice D is wrong because Maria has more, not less sugar.

Convert $$\frac{7}{11}$$ to a percentage: $$\frac{7}{11} = 0.636363... = 63.\overline{63}%$$. This is approximately $$63.64%$$. Among the choices, this falls between $$60%$$ and $$65%$$. Choice B is too narrow a range, though $$63.64%$$ does fall between $$63%$$ and $$64%$$, the question asks for benchmark percentages and choice A provides the appropriate broader range. Choice C ($$64%$$ to $$65%$$) doesn't contain $$63.64%$$. Choice D is too high.

Calculate each: A) $$\frac{2}{3} \times 0.75 = \frac{2}{3} \times \frac{3}{4} = \frac{1}{2} = 0.5$$. B) $$0.75 \times \frac{2}{3} = \frac{3}{4} \times \frac{2}{3} = \frac{1}{2} = 0.5$$. C) $$0.75 + \frac{2}{3} \times 0.75 = 0.75(1 + \frac{2}{3}) = 0.75 \times \frac{5}{3} = \frac{5}{4} = 1.25$$. D) $$\frac{2}{3} + 0.75 \times \frac{2}{3} = \frac{2}{3}(1 + 0.75) = \frac{2}{3} \times 1.75 = \frac{7}{6} \approx 1.167$$. Choice C has the greatest value at $$1.25$$.

When you encounter questions about equivalent representations of fractions and percentages, you need to verify whether different forms truly represent the same value through precise conversion.

To determine if $$\frac{2}{9}$$ equals $$22.\overline{2}%$$, convert the fraction to a decimal by dividing: $$2 \div 9 = 0.222...$$ or $$0.\overline{2}$$. To express this as a percentage, multiply by 100: $$0.\overline{2} \times 100 = 22.\overline{2}%$$. The notation $$22.\overline{2}%$$ means the digit 2 repeats infinitely (22.2222...%), which is exactly what we get from $$\frac{2}{9}$$. Therefore, the store's calculation is mathematically correct.

Choice A is wrong because $$\frac{2}{9}$$ absolutely can be expressed as a repeating decimal percentage—we just demonstrated this. Choice B contains a critical misunderstanding: $$22.22%$$ (which terminates) is actually an approximation, while $$22.\overline{2}%$$ (which repeats infinitely) is the exact value. The store used the precise notation, not an approximation. Choice C suggests $$\frac{200}{9}%$$, but this equals $$22.\overline{2}$$, which would mean $$22.\overline{2}%$$ of a percent—a much smaller value than intended.

The correct answer is D because both representations are mathematically equivalent.

Study tip: Master the bar notation for repeating decimals ($$\overline{2}$$ means the 2 repeats forever). This notation often appears on the SSAT to test precision versus approximation. Always convert fully rather than rounding when checking equivalencies.

A number that is $$25%$$ larger than $$\frac{4}{5}$$ equals $$\frac{4}{5} + 0.25 \times \frac{4}{5}$$. This can be written as $$\frac{4}{5}(1 + 0.25) = 1.25 \times \frac{4}{5}$$ (choice B). Since $$25% = \frac{1}{4}$$, it can also be written as $$\frac{4}{5} + \frac{1}{4} \times \frac{4}{5}$$ (choice C). Choice A incorrectly adds $$0.25$$ instead of $$25%$$ of $$\frac{4}{5}$$. Both B and C are mathematically equivalent and correct.