SSAT Upper Level Quiz: Comparing Rational Forms
9 questions · exam conditions
0:00
Comparing Rational FormsQuestion 1 of 9

A student claims that 37>0.42\frac{3}{7} > 0.42 because 37=0.428571...\frac{3}{7} = 0.428571... and 0.428571>0.420.428571 > 0.42. What is wrong with this reasoning?

The student's conversion is correct, but 0.428571...<0.430.428571... < 0.43, so more precision is needed
The student made an error; 37=0.42857142‾\frac{3}{7} = 0.42\overline{857142}, not 0.4285710.428571
The student correctly converted 37\frac{3}{7} to decimal form and the comparison is valid
The reasoning is flawed because 37\frac{3}{7} cannot be compared directly to 0.420.42
← Back to quizzes

SSAT Upper Level Quiz

SSAT Upper Level Quiz: Comparing Rational Forms

Practice Comparing Rational Forms in SSAT Upper Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Comparing Rational Forms, giving you a quick way to practice the rules, question types, and explanations that matter most for SSAT Upper Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A student claims that 37>0.42\frac{3}{7} > 0.42 because 37=0.428571...\frac{3}{7} = 0.428571... and 0.428571>0.420.428571 > 0.42. What is wrong with this reasoning?

  1. The student's conversion is correct, but 0.428571...<0.430.428571... < 0.43, so more precision is needed
  2. The student made an error; 37=0.42857142‾\frac{3}{7} = 0.42\overline{857142}, not 0.4285710.428571
  3. The student correctly converted 37\frac{3}{7} to decimal form and the comparison is valid (correct answer)
  4. The reasoning is flawed because 37\frac{3}{7} cannot be compared directly to 0.420.42
Explanation: 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 37\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 37>0.42\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 0.42857142‾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.

Question 2

The value 711\frac{7}{11} lies between which pair of benchmark percentages?

  1. Between 60%60\% and 65%65\% (correct answer)
  2. Between 63%63\% and 64%64\%
  3. Between 64%64\% and 65%65\%
  4. Between 65%65\% and 70%70\%
Explanation: Convert 711\frac{7}{11} to a percentage: 711=0.636363...=63.63‾%\frac{7}{11} = 0.636363... = 63.\overline{63}\%. This is approximately 63.64%63.64\%. Among the choices, this falls between 60%60\% and 65%65\%. Choice B is too narrow a range, though 63.64%63.64\% does fall between 63%63\% and 64%64\%, the question asks for benchmark percentages and choice A provides the appropriate broader range. Choice C (64%64\% to 65%65\%) doesn't contain 63.64%63.64\%. Choice D is too high.

Question 3

Which expression has the greatest value?

  1. 23\frac{2}{3} of 75%75\%
  2. 75%75\% of 23\frac{2}{3}
  3. 0.750.75 increased by 23\frac{2}{3} of itself (correct answer)
  4. 23\frac{2}{3} increased by 75%75\% of itself
Explanation: Calculate each: A) 23×0.75=23×34=12=0.5\frac{2}{3} \times 0.75 = \frac{2}{3} \times \frac{3}{4} = \frac{1}{2} = 0.5. B) 0.75×23=34×23=12=0.50.75 \times \frac{2}{3} = \frac{3}{4} \times \frac{2}{3} = \frac{1}{2} = 0.5. C) 0.75+23×0.75=0.75(1+23)=0.75×53=54=1.250.75 + \frac{2}{3} \times 0.75 = 0.75(1 + \frac{2}{3}) = 0.75 \times \frac{5}{3} = \frac{5}{4} = 1.25. D) 23+0.75×23=23(1+0.75)=23×1.75=76≈1.167\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 1.251.25.

Question 4

Which of the following lists the values 58\frac{5}{8}, 62%62\%, and 0.6270.627 in ascending order?

  1. 58<62%<0.627\frac{5}{8} < 62\% < 0.627
  2. 62%<58<0.62762\% < \frac{5}{8} < 0.627 (correct answer)
  3. 0.627<58<62%0.627 < \frac{5}{8} < 62\%
  4. 58<0.627<62%\frac{5}{8} < 0.627 < 62\%
Explanation: Convert all to decimals: 58=0.625\frac{5}{8} = 0.625, 62%=0.6262\% = 0.62, and 0.6270.627 stays as is. Ordering from least to greatest: 0.62<0.625<0.6270.62 < 0.625 < 0.627, which corresponds to 62%<58<0.62762\% < \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.

Question 5

If x=0.45‾x = 0.4\overline{5} and y=4190y = \frac{41}{90}, which statement about xx and yy is true?

  1. x>yx > y because repeating decimals are always greater than fractions
  2. x=yx = y because both represent the same rational number (correct answer)
  3. x<yx < y because 0.45‾<0.4560.4\overline{5} < 0.456 and 4190>0.456\frac{41}{90} > 0.456
  4. x<yx < y because the decimal representation is always less than the fraction
Explanation: Convert x=0.45‾x = 0.4\overline{5} to a fraction: Let x=0.4555...x = 0.4555... Then 10x=4.555...10x = 4.555... and 100x=45.555...100x = 45.555... So 100x−10x=45.555...−4.555...=41100x - 10x = 45.555... - 4.555... = 41, giving 90x=4190x = 41, so x=4190x = \frac{41}{90}. Therefore x=yx = y. Choices A and D make false generalizations about the relationship between decimals and fractions. Choice C incorrectly assumes 0.45‾<0.4560.4\overline{5} < 0.456, when actually 0.45‾=0.4555...>0.4560.4\overline{5} = 0.4555... > 0.456.

Question 6

Three students calculated the same expression and got 712\frac{7}{12}, 5813%58\frac{1}{3}\%, and 0.5833‾0.583\overline{3}. How many of these answers are correct?

  1. All three answers are correct and equivalent (correct answer)
  2. Only two of the answers are correct and equivalent
  3. Only one of the answers is correct
  4. All three answers are different values
Explanation: Convert all to the same form to compare: 712=0.583‾=0.5833‾\frac{7}{12} = 0.58\overline{3} = 0.583\overline{3}. To convert to percent: 712×100%=70012%=58412%=5813%\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.

Question 7

A recipe calls for 38\frac{3}{8} cup of sugar. Maria has 0.40.4 cups of sugar available. Which statement is true?

  1. Maria has exactly enough sugar for the recipe
  2. Maria has 140\frac{1}{40} cup more sugar than needed (correct answer)
  3. Maria has 0.0250.025 cups more sugar than needed
  4. Maria has 18\frac{1}{8} cup less sugar than needed
Explanation: First, convert 38\frac{3}{8} to decimal: 38=0.375\frac{3}{8} = 0.375. Maria has 0.40.4 cups, so she has 0.4−0.375=0.0250.4 - 0.375 = 0.025 cups extra. Converting 0.0250.025 to a fraction: 0.025=251000=1400.025 = \frac{25}{1000} = \frac{1}{40}. Choice A is wrong because 0.4≠0.3750.4 \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.

Question 8

A store offers a discount that reduces the original price by 29\frac{2}{9}. If this is equivalent to a 22.2‾%22.\overline{2}\% discount, what can you conclude?

  1. Both calculations are wrong; 29\frac{2}{9} cannot be expressed as a repeating decimal percentage
  2. The store made an error; 29\frac{2}{9} equals approximately 22.22%22.22\%, not 22.2‾%22.\overline{2}\%
  3. The store made an error; the percentage should be 2009%\frac{200}{9}\%
  4. The store's calculation is correct since both represent the same discount (correct answer)
Explanation: 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 29\frac{2}{9} equals 22.2‾%22.\overline{2}\%, convert the fraction to a decimal by dividing: 2÷9=0.222...2 \div 9 = 0.222... or 0.2‾0.\overline{2}. To express this as a percentage, multiply by 100: 0.2‾×100=22.2‾%0.\overline{2} \times 100 = 22.\overline{2}\%. The notation 22.2‾%22.\overline{2}\% means the digit 2 repeats infinitely (22.2222...%), which is exactly what we get from 29\frac{2}{9}. Therefore, the store's calculation is mathematically correct. Choice A is wrong because 29\frac{2}{9} absolutely can be expressed as a repeating decimal percentage—we just demonstrated this. Choice B contains a critical misunderstanding: 22.22%22.22\% (which terminates) is actually an approximation, while 22.2‾%22.\overline{2}\% (which repeats infinitely) is the exact value. The store used the precise notation, not an approximation. Choice C suggests 2009%\frac{200}{9}\%, but this equals 22.2‾22.\overline{2}, which would mean 22.2‾%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 (2‾\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.

Question 9

A number is 25%25\% larger than 45\frac{4}{5}. Which expression represents this number?

  1. 45+0.25\frac{4}{5} + 0.25
  2. 1.25×451.25 \times \frac{4}{5}
  3. 45+14×45\frac{4}{5} + \frac{1}{4} \times \frac{4}{5}
  4. Both B and C represent the same number (correct answer)
Explanation: A number that is 25%25\% larger than 45\frac{4}{5} equals 45+0.25×45\frac{4}{5} + 0.25 \times \frac{4}{5}. This can be written as 45(1+0.25)=1.25×45\frac{4}{5}(1 + 0.25) = 1.25 \times \frac{4}{5} (choice B). Since 25%=1425\% = \frac{1}{4}, it can also be written as 45+14×45\frac{4}{5} + \frac{1}{4} \times \frac{4}{5} (choice C). Choice A incorrectly adds 0.250.25 instead of 25%25\% of 45\frac{4}{5}. Both B and C are mathematically equivalent and correct.