Comparing Rational Forms - SSAT Upper Level Quantitative

1. Which of the following lists the values $\frac{5}{8}$, $62\%$, and $0.627$ in ascending order?
   Answer: $62\% < \frac{5}{8} < 0.627$ 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.
2. Which expression has the greatest value?
   Answer: $0.75$ increased by $\frac{2}{3}$ of itself 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 $1.25$.
3. The value $\frac{7}{11}$ lies between which pair of benchmark percentages?
   Answer: Between $60\%$ and $65\%$ Divide: $\frac{7}{11} = 0.636363\ldots = 63.\overline{63}\%$, or about $63.64\%$. This value is greater than $60\%$ and less than $65\%$, so it lies between $60\%$ and $65\%$. The range from $62\%$ to $63\%$ is too low, because $63.64\%$ is already above $63\%$; it catches students who truncate 63.64 to 63 and look below it. The range from $64\%$ to $65\%$ is too high, because $63.64\%$ is below $64\%$; this trap comes from rounding 63.64 up to 64 too early. The range from $65\%$ to $70\%$ is higher still. Strategy tip: Convert the fraction to a decimal with at least two decimal places before comparing it with percentages, and do not round until the end.
4. If $x = 0.4\overline{5}$ and $y = \frac{41}{90}$, which statement about $x$ and $y$ is true?
   Answer: $x = y$ because both represent the same rational number 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 $0.4\overline{5} = 0.4555... > 0.456$.
5. A store offers a discount that reduces the original price by $\frac{2}{9}$. If this is equivalent to a $22.\overline{2}\%$ discount, what can you conclude?
   Answer: The store's calculation is correct since both represent the same discount 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.
6. A student claims that $\frac{3}{7} > 0.42$ because $\frac{3}{7} = 0.428571...$ and $0.428571 > 0.42$. What is wrong with this reasoning?
   Answer: The student correctly converted $\frac{3}{7}$ to decimal form and the comparison is valid 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.
7. Three students calculated the same expression and got $\frac{7}{12}$, $58\frac{1}{3}\%$, and $0.583\overline{3}$. How many of these answers are correct?
   Answer: All three answers are correct and equivalent 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.
8. A recipe calls for $\frac{3}{8}$ cup of sugar. Maria has $0.4$ cups of sugar available. Which statement is true?
   Answer: Maria has $\frac{1}{40}$ cup more sugar than needed Convert the recipe amount to a decimal: $\frac{3}{8} = 0.375$ cup. Maria has $0.4$ cups, so she has $0.4 - 0.375 = 0.025$ cup more than she needs. As a fraction, $0.025 = \frac{25}{1000} = \frac{1}{40}$, so she has $\frac{1}{40}$ cup more sugar than needed. $0.25$ cups comes from misplacing the decimal point in the difference; it is ten times too large. "Exactly enough" is wrong because $0.4 \neq 0.375$. "$\frac{1}{8}$ cup less" has the direction wrong, since $0.4$ is greater than $0.375$, and its size is wrong as well. Strategy tip: To compare a fraction and a decimal, convert both to the same form, then subtract. Writing $0.4$ as $0.400$ keeps the place values lined up.
9. A number is $25\%$ larger than $\frac{4}{5}$. Which expression represents this number?
   Answer: Both B and C represent the same number 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.

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