ACCUPLACER Arithmetic Quiz: Comparing And Ordering Fractions
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Comparing And Ordering FractionsQuestion 1 of 20

A project requires a hole that is slightly larger than 58\frac{5}{8} inch but smaller than 23\frac{2}{3} inch. Which of the following drill bit sizes could be used?

35\frac{3}{5} inch
59\frac{5}{9} inch
1116\frac{11}{16} inch
1320\frac{13}{20} inch
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ACCUPLACER Arithmetic Quiz

ACCUPLACER Arithmetic Quiz: Comparing And Ordering Fractions

Practice Comparing And Ordering Fractions in ACCUPLACER Arithmetic 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 And Ordering Fractions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Arithmetic.

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 project requires a hole that is slightly larger than 58\frac{5}{8} inch but smaller than 23\frac{2}{3} inch. Which of the following drill bit sizes could be used?

  1. 35\frac{3}{5} inch
  2. 59\frac{5}{9} inch
  3. 1116\frac{11}{16} inch
  4. 1320\frac{13}{20} inch (correct answer)
Explanation: We need to find a fraction that is between 58\frac{5}{8} and 23\frac{2}{3}. Converting the given fractions to decimals gives a range: 58=0.625\frac{5}{8} = 0.625 and 23≈0.667\frac{2}{3} \approx 0.667. Now we check the answer choices: A) 35=0.6\frac{3}{5} = 0.6, which is too small. B) 59≈0.556\frac{5}{9} \approx 0.556, which is too small. C) 1116=0.6875\frac{11}{16} = 0.6875, which is too large. D) 1320=0.65\frac{13}{20} = 0.65, which is between 0.625 and 0.667. Thus, 1320\frac{13}{20} inch is the correct size.

Question 2

A pie is cut into 11 equal slices, and John eats 3 of them. A second, identical pie is cut into 9 equal slices, and Mary eats 3 of them. Which statement is true?

  1. John ate more pie than Mary.
  2. Mary ate more pie than John. (correct answer)
  3. They both ate the same amount of pie.
  4. Who ate more depends on the size of the pie.
Explanation: This problem asks us to compare the fractions of pie eaten. John ate 311\frac{3}{11} of a pie. Mary ate 39\frac{3}{9} of a pie. We need to compare 311\frac{3}{11} and 39\frac{3}{9}. When two fractions have the same numerator, the one with the smaller denominator is the larger fraction. Since 9 is smaller than 11, 39\frac{3}{9} is greater than 311\frac{3}{11}. Therefore, Mary ate more pie than John. Distractor D is incorrect because the problem states the pies are identical.

Question 3

In an election, Maria received 512\frac{5}{12} of the votes and David received 38\frac{3}{8} of the votes. Which of the following fractions represents a share of votes greater than David's share but less than Maria's share?

  1. 13\frac{1}{3}
  2. 25\frac{2}{5} (correct answer)
  3. 716\frac{7}{16}
  4. 49\frac{4}{9}
Explanation: First, compare the shares of Maria and David. Maria's share is 512\frac{5}{12} and David's is 38\frac{3}{8}. In decimal form, 512≈0.4167\frac{5}{12} \approx 0.4167 and 38=0.375\frac{3}{8} = 0.375. We need to find a fraction that is between 0.375 and 0.4167. Now check the options: A) 13≈0.333\frac{1}{3} \approx 0.333 (too small). B) 25=0.4\frac{2}{5} = 0.4 (this is between 0.375 and 0.4167). C) 716=0.4375\frac{7}{16} = 0.4375 (too large). D) 49≈0.444\frac{4}{9} \approx 0.444 (too large). Thus, 25\frac{2}{5} is the correct answer.

Question 4

A recipe for soup requires 23\frac{2}{3} cup of stock, and a recipe for a sauce requires 34\frac{3}{4} cup of stock. If you have two separate 1-cup containers of stock and use one for each recipe, which recipe leaves a greater amount of stock remaining in its container?

  1. The soup recipe leaves a greater amount. (correct answer)
  2. The sauce recipe leaves a greater amount.
  3. Both recipes leave the same amount remaining.
  4. The result depends on the initial amount of stock.
Explanation: First, calculate the amount of stock remaining for each recipe. For the soup: 1−23=131 - \frac{2}{3} = \frac{1}{3} cup remains. For the sauce: 1−34=141 - \frac{3}{4} = \frac{1}{4} cup remains. Now, compare the remaining amounts, 13\frac{1}{3} and 14\frac{1}{4}. Since 13\frac{1}{3} is greater than 14\frac{1}{4}, the soup recipe leaves a greater amount of stock remaining. A common error is to compare the initial amounts used (23\frac{2}{3} and 34\frac{3}{4}) instead of the remainders.

Question 5

A water tank is 38\frac{3}{8} full. After some water is used, it is 13\frac{1}{3} full. Later, it is filled until it is 34\frac{3}{4} full. Which list shows these three fractional amounts of water from least to greatest?

  1. 13,38,34\frac{1}{3}, \frac{3}{8}, \frac{3}{4} (correct answer)
  2. 13,34,38\frac{1}{3}, \frac{3}{4}, \frac{3}{8}
  3. 38,13,34\frac{3}{8}, \frac{1}{3}, \frac{3}{4}
  4. 34,38,13\frac{3}{4}, \frac{3}{8}, \frac{1}{3}
Explanation: The problem asks to order the fractions 38\frac{3}{8}, 13\frac{1}{3}, and 34\frac{3}{4}. To compare them, we can find a common denominator, which is 24. Converting the fractions: 38=924\frac{3}{8} = \frac{9}{24}, 13=824\frac{1}{3} = \frac{8}{24}, and 34=1824\frac{3}{4} = \frac{18}{24}. By comparing the numerators (8, 9, 18), the order from least to greatest is 824,924,1824\frac{8}{24}, \frac{9}{24}, \frac{18}{24}, which corresponds to 13,38,34\frac{1}{3}, \frac{3}{8}, \frac{3}{4}.

Question 6

Three runners are practicing on a track. Ali ran 78\frac{7}{8} of a mile, Ben ran 56\frac{5}{6} of a mile, and Carmen ran 1315\frac{13}{15} of a mile. Which statement correctly compares the distances they ran?

  1. Ben ran the farthest distance.
  2. Carmen ran a shorter distance than Ben.
  3. Ali ran the farthest distance. (correct answer)
  4. All three runners ran the same distance.
Explanation: To compare the distances, we compare the fractions 78\frac{7}{8}, 56\frac{5}{6}, and 1315\frac{13}{15}. Converting to decimals is an efficient method: Ali: 78=0.875\frac{7}{8} = 0.875. Ben: 56≈0.833\frac{5}{6} \approx 0.833. Carmen: 1315≈0.867\frac{13}{15} \approx 0.867. Comparing these decimals, we see that 0.875>0.867>0.8330.875 > 0.867 > 0.833. Therefore, Ali ran the farthest distance. Distractor A is wrong because Ben ran the shortest distance. Distractor B is wrong because Carmen (0.867) ran farther than Ben (0.833).

Question 7

Four students are reading the same book. Amy has read 58\frac{5}{8} of the book, Ben has read 23\frac{2}{3} of the book, Chloe has read 35\frac{3}{5} of the book, and David has read 710\frac{7}{10} of the book. Who has read the most?

  1. Amy
  2. Ben
  3. Chloe
  4. David (correct answer)
Explanation: To find out who has read the most, we need to find the largest fraction among 58\frac{5}{8}, 23\frac{2}{3}, 35\frac{3}{5}, and 710\frac{7}{10}. Converting to decimals is an efficient way to compare: Amy (58\frac{5}{8}) = 0.625. Ben (23\frac{2}{3}) ≈ 0.667. Chloe (35\frac{3}{5}) = 0.60. David (710\frac{7}{10}) = 0.70. The largest decimal is 0.70, which corresponds to the fraction read by David. Therefore, David has read the most.

Question 8

A recipe is scaled down so that 23\frac{2}{3} of the original amount of each ingredient is used. If the original recipe called for 34\frac{3}{4} cup of milk and 56\frac{5}{6} cup of flour, which ingredient quantity is larger in the scaled recipe?

  1. Milk is larger by 118\frac{1}{18} cup
  2. Flour is larger by 118\frac{1}{18} cup (correct answer)
  3. Milk is larger by 19\frac{1}{9} cup
  4. Flour is larger by 19\frac{1}{9} cup
Explanation: Scale each ingredient by 23\frac{2}{3}: Milk: 23×34=612=12\frac{2}{3} × \frac{3}{4} = \frac{6}{12} = \frac{1}{2} cup. Flour: 23×56=1018=59\frac{2}{3} × \frac{5}{6} = \frac{10}{18} = \frac{5}{9} cup. To compare 12\frac{1}{2} and 59\frac{5}{9}, convert to common denominator 18: 12=918\frac{1}{2} = \frac{9}{18} and 59=1018\frac{5}{9} = \frac{10}{18}. Since 1018>918\frac{10}{18} > \frac{9}{18}, flour is larger. The difference is 1018−918=118\frac{10}{18} - \frac{9}{18} = \frac{1}{18} cup. Choice A reverses which ingredient is larger. Choice C and D use 19\frac{1}{9} as the difference, which would be correct if comparing 1018\frac{10}{18} and 818\frac{8}{18}, but that's not our calculation.

Question 9

A recipe calls for 23\frac{2}{3} cup of flour, but Maria only has a 14\frac{1}{4} cup measuring cup. She fills it completely 3 times. What fraction represents how much more flour Maria has compared to what the recipe requires?

  1. 112\frac{1}{12} (correct answer)
  2. 512\frac{5}{12}
  3. 712\frac{7}{12}
  4. 1112\frac{11}{12}
Explanation: Maria has 3×14=343 \times \frac{1}{4} = \frac{3}{4} cup of flour. The recipe needs 23\frac{2}{3} cup. To compare: 34−23=912−812=112\frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12}. Maria has 112\frac{1}{12} cup more than needed. Choice B incorrectly subtracts 23−14\frac{2}{3} - \frac{1}{4}. Choice C adds the fractions instead of subtracting. Choice D represents the total amount Maria has after converting to twelfths.

Question 10

Three friends are comparing their test scores as fractions of total points possible. Alex scored 1720\frac{17}{20}, Beth scored 2125\frac{21}{25}, and Carlos scored 3340\frac{33}{40}. If they want to rank themselves from lowest to highest score, what is the correct order?

  1. Alex, Carlos, Beth (lowest to highest)
  2. Carlos, Alex, Beth (lowest to highest) (correct answer)
  3. Carlos, Beth, Alex (lowest to highest)
  4. Beth, Alex, Carlos (lowest to highest)
Explanation: Convert all fractions to a common denominator of 200: Alex: 1720=170200\frac{17}{20} = \frac{170}{200}, Beth: 2125=168200\frac{21}{25} = \frac{168}{200}, Carlos: 3340=165200\frac{33}{40} = \frac{165}{200}. From lowest to highest: 165200<168200<170200\frac{165}{200} < \frac{168}{200} < \frac{170}{200}, so Carlos < Beth < Alex. Choice A incorrectly places Alex as lowest. Choice C reverses Beth and Alex. Choice D completely reverses the order by putting Beth first.

Question 11

Which of the following lists of fractions is ordered from greatest to least?

  1. 411,49,47,45\frac{4}{11}, \frac{4}{9}, \frac{4}{7}, \frac{4}{5}
  2. 45,47,49,411\frac{4}{5}, \frac{4}{7}, \frac{4}{9}, \frac{4}{11} (correct answer)
  3. 45,49,47,411\frac{4}{5}, \frac{4}{9}, \frac{4}{7}, \frac{4}{11}
  4. 411,45,47,49\frac{4}{11}, \frac{4}{5}, \frac{4}{7}, \frac{4}{9}
Explanation: When comparing fractions that have the same numerator, the fraction with the smallest denominator has the greatest value. The numerators are all 4. The denominators are 5, 7, 9, and 11. Ordering the denominators from least to greatest gives 5, 7, 9, 11. Therefore, the corresponding fractions ordered from greatest to least are 45,47,49,411\frac{4}{5}, \frac{4}{7}, \frac{4}{9}, \frac{4}{11}. Distractor A incorrectly lists them from least to greatest.

Question 12

A city's budget is spent as follows: 25\frac{2}{5} on public safety, 14\frac{1}{4} on education, and 310\frac{3}{10} on infrastructure. Which statement accurately describes these expenditures?

  1. Education spending is greater than infrastructure spending.
  2. Infrastructure spending is the largest of the three categories.
  3. Public safety spending is greater than education spending. (correct answer)
  4. Education spending is the largest of the three categories.
Explanation: To compare the fractions, find a common denominator, which for 5, 4, and 10 is 20. Public safety: 25=820\frac{2}{5} = \frac{8}{20}. Education: 14=520\frac{1}{4} = \frac{5}{20}. Infrastructure: 310=620\frac{3}{10} = \frac{6}{20}. Now compare the numerators. Public safety (8) > Infrastructure (6) > Education (5). Let's evaluate the statements: A is false (5 < 6). B is false (8 > 6). C is true (8 > 5). D is false (5 is the smallest). Therefore, the only true statement is that public safety spending is greater than education spending.

Question 13

Which of the following fractions has the greatest value?

  1. 78\frac{7}{8}
  2. 1516\frac{15}{16}
  3. 1920\frac{19}{20}
  4. 3940\frac{39}{40} (correct answer)
Explanation: Each of these fractions is one part less than a whole (1). We can compare them by seeing how far each is from 1. A) 1−78=181 - \frac{7}{8} = \frac{1}{8}. B) 1−1516=1161 - \frac{15}{16} = \frac{1}{16}. C) 1−1920=1201 - \frac{19}{20} = \frac{1}{20}. D) 1−3940=1401 - \frac{39}{40} = \frac{1}{40}. The fraction that is 'missing' the smallest piece to make a whole is the largest fraction. Since 140\frac{1}{40} is the smallest of these missing pieces, 3940\frac{39}{40} is the fraction with the greatest value.

Question 14

Which of the following fractions is the largest?

  1. 1120\frac{11}{20}
  2. 47\frac{4}{7}
  3. 58\frac{5}{8}
  4. 1827\frac{18}{27} (correct answer)
Explanation: To find the largest fraction, we can compare them. Notice that the fraction 1827\frac{18}{27} can be simplified. Both 18 and 27 are divisible by 9, so 1827=23\frac{18}{27} = \frac{2}{3}. Now compare 23\frac{2}{3} with the other fractions by converting to decimals: A) 1120=0.55\frac{11}{20} = 0.55. B) 47≈0.571\frac{4}{7} \approx 0.571. C) 58=0.625\frac{5}{8} = 0.625. D) 1827=23≈0.667\frac{18}{27} = \frac{2}{3} \approx 0.667. Since 0.667 is the largest decimal value, 1827\frac{18}{27} is the largest fraction.

Question 15

Which of the following lists the fractions 35\frac{3}{5}, 58\frac{5}{8}, and 23\frac{2}{3} in order from least to greatest?

  1. 35,58,23\frac{3}{5}, \frac{5}{8}, \frac{2}{3} (correct answer)
  2. 23,58,35\frac{2}{3}, \frac{5}{8}, \frac{3}{5}
  3. 58,35,23\frac{5}{8}, \frac{3}{5}, \frac{2}{3}
  4. 35,23,58\frac{3}{5}, \frac{2}{3}, \frac{5}{8}
Explanation: To compare the fractions, find a common denominator or convert them to decimals. The least common multiple of 5, 8, and 3 is 120. Convert each fraction: 35=72120\frac{3}{5} = \frac{72}{120}, 58=75120\frac{5}{8} = \frac{75}{120}, and 23=80120\frac{2}{3} = \frac{80}{120}. Comparing the numerators (72, 75, 80) gives the order 35\frac{3}{5}, 58\frac{5}{8}, 23\frac{2}{3}. Alternatively, converting to decimals: 35=0.6\frac{3}{5} = 0.6, 58=0.625\frac{5}{8} = 0.625, and 23≈0.667\frac{2}{3} \approx 0.667. The correct order from least to greatest is 35,58,23\frac{3}{5}, \frac{5}{8}, \frac{2}{3}.

Question 16

A number nn satisfies the inequality 25<n<23\frac{2}{5} < n < \frac{2}{3}. Which of the following could be the value of nn?

  1. 13\frac{1}{3}
  2. 38\frac{3}{8}
  3. 58\frac{5}{8} (correct answer)
  4. 710\frac{7}{10}
Explanation: First, establish the range in decimal form for easier comparison. 25=0.4\frac{2}{5} = 0.4 and 23≈0.667\frac{2}{3} \approx 0.667. We need to find the option that falls between 0.4 and 0.667. A) 13≈0.333\frac{1}{3} \approx 0.333 (too small). B) 38=0.375\frac{3}{8} = 0.375 (too small). C) 58=0.625\frac{5}{8} = 0.625 (this is between 0.4 and 0.667). D) 710=0.7\frac{7}{10} = 0.7 (too large). Therefore, 58\frac{5}{8} is a possible value for nn.

Question 17

Which of the following fractions is greater than 34\frac{3}{4} but less than 45\frac{4}{5}?

  1. 710\frac{7}{10}
  2. 1115\frac{11}{15}
  3. 1316\frac{13}{16}
  4. 3140\frac{31}{40} (correct answer)
Explanation: To solve this, find a common denominator or convert the fractions to decimals. As decimals, 34=0.75\frac{3}{4} = 0.75 and 45=0.80\frac{4}{5} = 0.80. We need to find a fraction between 0.75 and 0.80. Let's convert the options to decimals: A) 710=0.70\frac{7}{10} = 0.70 (too small). B) 1115≈0.733\frac{11}{15} \approx 0.733 (too small). C) 1316=0.8125\frac{13}{16} = 0.8125 (too large). D) 3140=0.775\frac{31}{40} = 0.775 (which is between 0.75 and 0.80). Therefore, 3140\frac{31}{40} is the correct answer.

Question 18

Which of the following inequalities is true?

  1. −23>−35-\frac{2}{3} > -\frac{3}{5}
  2. −78<−89-\frac{7}{8} < -\frac{8}{9}
  3. −14>−15-\frac{1}{4} > -\frac{1}{5}
  4. −512<−38-\frac{5}{12} < -\frac{3}{8} (correct answer)
Explanation: To compare negative fractions, remember that the fraction with the smaller absolute value is the greater number (closer to zero). Let's convert to decimals or use a common denominator. For D: −512-\frac{5}{12} and −38-\frac{3}{8}. The common denominator is 24. −512=−1024-\frac{5}{12} = -\frac{10}{24} and −38=−924-\frac{3}{8} = -\frac{9}{24}. Since -10 is less than -9, −1024<−924-\frac{10}{24} < -\frac{9}{24}, which means −512<−38-\frac{5}{12} < -\frac{3}{8} is true. For A: −23≈−0.667-\frac{2}{3} \approx -0.667 and −35=−0.6-\frac{3}{5} = -0.6; −0.667<−0.6-0.667 < -0.6. For B: 78<89\frac{7}{8} < \frac{8}{9}, so −78>−89-\frac{7}{8} > -\frac{8}{9}. For C: 14>15\frac{1}{4} > \frac{1}{5}, so −14<−15-\frac{1}{4} < -\frac{1}{5}.

Question 19

Three of the following fractions are equivalent to each other. Which fraction is NOT equivalent to the others?

  1. 912\frac{9}{12}
  2. 1520\frac{15}{20}
  3. 2128\frac{21}{28}
  4. 2436\frac{24}{36} (correct answer)
Explanation: To determine which fraction is not equivalent, we simplify each fraction to its lowest terms. A) 912\frac{9}{12} simplifies to 34\frac{3}{4} by dividing the numerator and denominator by 3. B) 1520\frac{15}{20} simplifies to 34\frac{3}{4} by dividing by 5. C) 2128\frac{21}{28} simplifies to 34\frac{3}{4} by dividing by 7. D) 2436\frac{24}{36} simplifies to 23\frac{2}{3} by dividing by 12. Since 23\frac{2}{3} is not equal to 34\frac{3}{4}, 2436\frac{24}{36} is the fraction that is not equivalent to the others.

Question 20

Which of the following lists the numbers 1141\frac{1}{4}, 65\frac{6}{5}, and 97\frac{9}{7} in order from least to greatest?

  1. 97,114,65\frac{9}{7}, 1\frac{1}{4}, \frac{6}{5}
  2. 114,65,971\frac{1}{4}, \frac{6}{5}, \frac{9}{7}
  3. 65,97,114\frac{6}{5}, \frac{9}{7}, 1\frac{1}{4}
  4. 65,114,97\frac{6}{5}, 1\frac{1}{4}, \frac{9}{7} (correct answer)
Explanation: To order the numbers, it's helpful to convert them to the same format, such as decimals. 114=1.251\frac{1}{4} = 1.25. 65=1.2\frac{6}{5} = 1.2. 97≈1.286\frac{9}{7} \approx 1.286. Now, order these decimal values from least to greatest: 1.2, 1.25, 1.286. This corresponds to the original numbers in the order 65,114,97\frac{6}{5}, 1\frac{1}{4}, \frac{9}{7}.