ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Comparing And Ordering Rational Numbers
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Comparing And Ordering Rational NumbersQuestion 1 of 20

Which of the following numbers is farthest from −23-\frac{2}{3} on the number line?

-2
−12-\frac{1}{2}
0
1
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Comparing And Ordering Rational Numbers

Practice Comparing And Ordering Rational Numbers in ACCUPLACER Quantitative Reasoning, Algebra & Statistics 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 Rational Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Quantitative Reasoning, Algebra & Statistics.

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.

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Question 1

Which of the following numbers is farthest from −23-\frac{2}{3} on the number line?

  1. -2
  2. −12-\frac{1}{2}
  3. 0
  4. 1 (correct answer)
Explanation: To find which number is farthest, we calculate the absolute distance between each choice and −23-\frac{2}{3}. A) Distance from -2: ∣−2−(−23)∣=∣−63+23∣=∣−43∣=43|-2 - (-\frac{2}{3})| = |-\frac{6}{3} + \frac{2}{3}| = |-\frac{4}{3}| = \frac{4}{3} B) Distance from −12-\frac{1}{2}: ∣−12−(−23)∣=∣−36+46∣=∣16∣=16|-\frac{1}{2} - (-\frac{2}{3})| = |-\frac{3}{6} + \frac{4}{6}| = |\frac{1}{6}| = \frac{1}{6} C) Distance from 0: ∣0−(−23)∣=∣23∣=23|0 - (-\frac{2}{3})| = |\frac{2}{3}| = \frac{2}{3} D) Distance from 1: ∣1−(−23)∣=∣33+23∣=∣53∣=53|1 - (-\frac{2}{3})| = |\frac{3}{3} + \frac{2}{3}| = |\frac{5}{3}| = \frac{5}{3} Comparing the distances 43,16,23,53\frac{4}{3}, \frac{1}{6}, \frac{2}{3}, \frac{5}{3}, the largest value is 53\frac{5}{3}. Therefore, 1 is the farthest from −23-\frac{2}{3}.

Question 2

Let x=−54x = -\frac{5}{4}. Which of the following lists the values xx, ∣x∣|x|, and x+2x+2 in order from least to greatest?

  1. x,x+2,∣x∣x, x+2, |x| (correct answer)
  2. x,∣x∣,x+2x, |x|, x+2
  3. x+2,x,∣x∣x+2, x, |x|
  4. ∣x∣,x+2,x|x|, x+2, x
Explanation: First, calculate the value of each expression given x=−54x = -\frac{5}{4}. x=−54=−1.25x = -\frac{5}{4} = -1.25 ∣x∣=∣−54∣=54=1.25|x| = |-\frac{5}{4}| = \frac{5}{4} = 1.25 x+2=−54+2=−54+84=34=0.75x+2 = -\frac{5}{4} + 2 = -\frac{5}{4} + \frac{8}{4} = \frac{3}{4} = 0.75 The three values are -1.25, 1.25, and 0.75. Ordering these from least to greatest gives: −1.25<0.75<1.25-1.25 < 0.75 < 1.25. This corresponds to the order x,x+2,∣x∣x, x+2, |x|.

Question 3

Three shelves are being installed. Shelf A has a length of 3383 \frac{3}{8} feet. Shelf B has a length of 3.43.4 feet. Shelf C has a length of 103\frac{10}{3} feet. Which of the following correctly orders the shelves from shortest to longest?

  1. A, B, C
  2. B, A, C
  3. C, A, B (correct answer)
  4. C, B, A
Explanation: To compare the lengths, convert all measurements to the same format, such as decimals. Shelf A: 338=3.3753 \frac{3}{8} = 3.375 feet. Shelf B: 3.43.4 feet. Shelf C: 103=3.333...\frac{10}{3} = 3.333... feet. Now, compare the decimal values: 3.333...<3.375<3.43.333... < 3.375 < 3.4. Therefore, the correct order from shortest to longest is C, A, B.

Question 4

A recipe requires 34\frac{3}{4} cup of flour. A baker has a scoop that holds 13\frac{1}{3} cup and another that holds 12\frac{1}{2} cup. The baker uses two scoops of the smaller size and one scoop of the larger size. Which statement accurately compares the amount of flour used to the amount required?

  1. The amount used is 512\frac{5}{12} cup less than required.
  2. The amount used is 112\frac{1}{12} cup less than required.
  3. The amount used is 112\frac{1}{12} cup more than required.
  4. The amount used is 512\frac{5}{12} cup more than required. (correct answer)
Explanation: First, calculate the total amount of flour used. Two scoops of 13\frac{1}{3} cup is 2×13=232 \times \frac{1}{3} = \frac{2}{3} cup. One scoop of 12\frac{1}{2} cup is 12\frac{1}{2} cup. The total amount used is 23+12\frac{2}{3} + \frac{1}{2}. To add these, find a common denominator, which is 6: 46+36=76\frac{4}{6} + \frac{3}{6} = \frac{7}{6} cup. Next, compare the amount used (76\frac{7}{6} cup) to the amount required (34\frac{3}{4} cup). Find a common denominator, which is 12. Amount used: 76=1412\frac{7}{6} = \frac{14}{12}. Amount required: 34=912\frac{3}{4} = \frac{9}{12}. The difference is 1412−912=512\frac{14}{12} - \frac{9}{12} = \frac{5}{12}. Since the amount used is greater than the amount required, the baker used 512\frac{5}{12} cup more than required.

Question 5

The ideal weight for a manufactured part is 78\frac{7}{8} ounces. A part is acceptable if its weight is within 132\frac{1}{32} ounce of the ideal weight. Which of the following weights would cause a part to be rejected?

  1. 1316\frac{13}{16} ounces (correct answer)
  2. 5564\frac{55}{64} ounces
  3. 78\frac{7}{8} ounces
  4. 2932\frac{29}{32} ounces
Explanation: First, find the acceptable range of weights. The ideal weight is 78\frac{7}{8} oz. Convert this to a fraction with a denominator of 32: 78=2832\frac{7}{8} = \frac{28}{32} oz. The acceptable range is 2832±132\frac{28}{32} \pm \frac{1}{32}, which is from 2732\frac{27}{32} oz to 2932\frac{29}{32} oz, inclusive. A part is rejected if its weight is outside this range. A) 1316=2632\frac{13}{16} = \frac{26}{32}. This weight is less than 2732\frac{27}{32}, so it is outside the acceptable range and would be rejected. B) 5564=27.532\frac{55}{64} = \frac{27.5}{32}. This is within the range [2732,2932][\frac{27}{32}, \frac{29}{32}], so it is acceptable. C) 78=2832\frac{7}{8} = \frac{28}{32}. This is the ideal weight, so it is acceptable. D) 2932\frac{29}{32}. This is the upper limit of the acceptable range, so it is acceptable.

Question 6

Three friends are running a race. After 10 minutes, Alex has completed 25\frac{2}{5} of the race, Ben has completed 38\frac{3}{8} of the race, and Carlos has completed 0.41 of the race. Who is in the lead, and who is in last place?

  1. Carlos is in the lead; Ben is in last place. (correct answer)
  2. Alex is in the lead; Ben is in last place.
  3. Carlos is in the lead; Alex is in last place.
  4. Ben is in the lead; Carlos is in last place.
Explanation: To determine the order, we need to compare the fractions of the race each person has completed. It's easiest to convert all numbers to decimals. Alex: 25=0.40\frac{2}{5} = 0.40 Ben: 38=0.375\frac{3}{8} = 0.375 Carlos: 0.410.41 Comparing the decimals: 0.41>0.40>0.3750.41 > 0.40 > 0.375. This means Carlos has completed the most of the race and is in the lead. Ben has completed the least and is in last place.

Question 7

Three pumps remove water from a tank at rates of 38\frac{3}{8} tank per hour, 0.40.4 tank per hour, and 720\frac{7}{20} tank per hour, respectively. If only one pump can operate at a time, which pump should be used to empty the tank fastest?

  1. The first pump should be used because 38\frac{3}{8} has the largest numerator
  2. The second pump should be used because 0.40.4 tank per hour is the fastest rate (correct answer)
  3. The third pump should be used because 720\frac{7}{20} tank per hour is the fastest rate
  4. The first pump should be used because 38\frac{3}{8} tank per hour is the fastest rate
Explanation: To find the fastest pump, convert all rates to the same form and compare. 38=0.375\frac{3}{8} = 0.375, 0.4=0.40.4 = 0.4, and 720=0.35\frac{7}{20} = 0.35. Comparing: 0.4>0.375>0.350.4 > 0.375 > 0.35. Therefore, the second pump with rate 0.40.4 tank per hour is fastest. Choice A is incorrect because having the largest numerator doesn't guarantee the largest fraction value. Choice C is incorrect because 720=0.35\frac{7}{20} = 0.35 is actually the slowest rate. Choice D is incorrect because 38=0.375\frac{3}{8} = 0.375 is not the fastest rate.

Question 8

If x=−23x = -\frac{2}{3}, which of the following expressions has the greatest value?

  1. xx
  2. x2x^2
  3. 1x\frac{1}{x}
  4. −x-x (correct answer)
Explanation: We evaluate each expression with x=−23x = -\frac{2}{3}. A) x=−23x = -\frac{2}{3} B) x2=(−23)2=49x^2 = (-\frac{2}{3})^2 = \frac{4}{9} C) 1x=1−2/3=−32\frac{1}{x} = \frac{1}{-2/3} = -\frac{3}{2} D) −x=−(−23)=23-x = -(-\frac{2}{3}) = \frac{2}{3} Now, we compare the values: −23≈−0.667-\frac{2}{3} \approx -0.667, 49≈0.444\frac{4}{9} \approx 0.444, −32=−1.5-\frac{3}{2} = -1.5, and 23≈0.667\frac{2}{3} \approx 0.667. The greatest value among these is 23\frac{2}{3}, which corresponds to the expression −x-x.

Question 9

For which of the following integer values of nn is the inequality 25<n12<34\frac{2}{5} < \frac{n}{12} < \frac{3}{4} true?

  1. 4
  2. 5 (correct answer)
  3. 9
  4. 10
Explanation: To solve the inequality 25<n12<34\frac{2}{5} < \frac{n}{12} < \frac{3}{4}, we can multiply all parts of the inequality by 12 to isolate nn. 12⋅25<12⋅n12<12⋅3412 \cdot \frac{2}{5} < 12 \cdot \frac{n}{12} < 12 \cdot \frac{3}{4} 245<n<9\frac{24}{5} < n < 9 Now, convert 245\frac{24}{5} to a decimal: 4.84.8. The inequality becomes 4.8<n<94.8 < n < 9. The integers nn that satisfy this condition are 5, 6, 7, and 8. Of the choices given, only n=5n=5 is in this range.

Question 10

What rational number is exactly halfway between −35-\frac{3}{5} and 710\frac{7}{10} on the number line?

  1. −110-\frac{1}{10}
  2. 120\frac{1}{20} (correct answer)
  3. 110\frac{1}{10}
  4. 1320\frac{13}{20}
Explanation: To find the number exactly halfway between two numbers, we calculate their average. First, express both numbers with a common denominator, which is 10. −35=−610-\frac{3}{5} = -\frac{6}{10}. The two numbers are −610-\frac{6}{10} and 710\frac{7}{10}. Average = −610+7102=1102=110×12=120\frac{-\frac{6}{10} + \frac{7}{10}}{2} = \frac{\frac{1}{10}}{2} = \frac{1}{10} \times \frac{1}{2} = \frac{1}{20}.

Question 11

Let x=3/42/5x = \frac{3/4}{2/5} and y=4/51/2y = \frac{4/5}{1/2}. Which of the following inequalities is true?

  1. x<yx < y
  2. x>yx > y (correct answer)
  3. x=yx = y
  4. y>2y > 2
Explanation: First, simplify the expressions for xx and yy. x=3/42/5=34×52=158x = \frac{3/4}{2/5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} y=4/51/2=45×21=85y = \frac{4/5}{1/2} = \frac{4}{5} \times \frac{2}{1} = \frac{8}{5} Now, compare x=158x = \frac{15}{8} and y=85y = \frac{8}{5}. Convert them to decimals: x=1.875x = 1.875 and y=1.6y = 1.6. Since 1.875>1.61.875 > 1.6, we have x>yx > y.

Question 12

The number kk is a rational number such that k>1k > 1. The number mm is a rational number such that 0<m<10 < m < 1. Which of the following inequalities must be true?

  1. k2<kk^2 < k
  2. m2>mm^2 > m
  3. k>m2k > m^2 (correct answer)
  4. m>k2m > k^2
Explanation: Let's analyze the properties of kk and mm. If k>1k > 1, squaring it results in a larger number. For example, if k=2k=2, k2=4k^2=4, so k2>kk^2 > k. This makes choice A false. If 0<m<10 < m < 1, squaring it results in a smaller number. For example, if m=0.5m=0.5, m2=0.25m^2=0.25, so m2<mm^2 < m. This makes choice B false. Now let's compare kk and m2m^2. We know k>1k > 1. We also know that if 0<m<10 < m < 1, then 0<m2<10 < m^2 < 1. Since kk is greater than 1 and m2m^2 is less than 1, it must be true that k>m2k > m^2. This makes choice C correct. For choice D, m>k2m > k^2 is false because mm is less than 1 and k2k^2 is greater than 1.

Question 13

Which of the following fractions has the greatest value?

  1. 1113\frac{11}{13}
  2. 1315\frac{13}{15}
  3. 1517\frac{15}{17}
  4. 1719\frac{17}{19} (correct answer)
Explanation: One way to compare these fractions is to consider how far each is from 1. A) 1−1113=2131 - \frac{11}{13} = \frac{2}{13} B) 1−1315=2151 - \frac{13}{15} = \frac{2}{15} C) 1−1517=2171 - \frac{15}{17} = \frac{2}{17} D) 1−1719=2191 - \frac{17}{19} = \frac{2}{19} The fraction with the greatest value will be the one that is closest to 1, meaning its difference from 1 is the smallest. We need to find the smallest fraction among 213,215,217,219\frac{2}{13}, \frac{2}{15}, \frac{2}{17}, \frac{2}{19}. Since all these fractions have the same numerator (2), the one with the largest denominator will be the smallest. The largest denominator is 19, so 219\frac{2}{19} is the smallest difference. Therefore, 1719\frac{17}{19} is the fraction closest to 1 and has the greatest value.

Question 14

A temperature reading was −45-\frac{4}{5}°C. An hour later, it was −56-\frac{5}{6}°C. Which of the following temperatures was recorded between these two readings?

  1. -0.9°C
  2. -0.85°C
  3. -0.82°C (correct answer)
  4. -0.79°C
Explanation: First, convert the fractional temperatures to decimals to establish the range. −45=−0.8-\frac{4}{5} = -0.8°C. −56≈−0.833-\frac{5}{6} \approx -0.833°C. Since −0.833<−0.8-0.833 < -0.8, the temperature dropped. We are looking for a temperature TT such that −56<T<−45-\frac{5}{6} < T < -\frac{4}{5}, or −0.833<T<−0.8-0.833 < T < -0.8. A) -0.9 is less than -0.833. B) -0.85 is less than -0.833. C) -0.82 is between -0.833 and -0.8. D) -0.79 is greater than -0.8. Therefore, -0.82°C is a temperature between the two readings.

Question 15

A point PP on a number line is located at −138-\frac{13}{8}. Which of the following points is closest to PP?

  1. -1.75
  2. -1.6 (correct answer)
  3. −117-\frac{11}{7}
  4. −53-\frac{5}{3}
Explanation: First, convert −138-\frac{13}{8} to a decimal: −13÷8=−1.625-13 \div 8 = -1.625. Now, find the distance from -1.625 to each of the given points. A) Distance to -1.75: ∣−1.75−(−1.625)∣=∣−0.125∣=0.125|-1.75 - (-1.625)| = |-0.125| = 0.125 B) Distance to -1.6: ∣−1.6−(−1.625)∣=∣0.025∣=0.025|-1.6 - (-1.625)| = |0.025| = 0.025 C) Distance to −117≈−1.571-\frac{11}{7} \approx -1.571: ∣−1.571−(−1.625)∣=∣0.054∣=0.054|-1.571 - (-1.625)| = |0.054| = 0.054 D) Distance to −53≈−1.667-\frac{5}{3} \approx -1.667: ∣−1.667−(−1.625)∣=∣−0.042∣=0.042|-1.667 - (-1.625)| = |-0.042| = 0.042 Comparing the distances (0.125, 0.025, 0.054, 0.042), the smallest distance is 0.025. Therefore, -1.6 is the closest point to PP.

Question 16

If a=−32a = -\frac{3}{2}, b=54b = \frac{5}{4}, and c=−75c = -\frac{7}{5}, which of the following is the correct ordering of aa, bb, and cc from least to greatest?

  1. a<c<ba < c < b (correct answer)
  2. c<a<bc < a < b
  3. a<b<ca < b < c
  4. b<c<ab < c < a
Explanation: To order the numbers, it is helpful to convert them to decimals. a=−32=−1.5a = -\frac{3}{2} = -1.5 b=54=1.25b = \frac{5}{4} = 1.25 c=−75=−1.4c = -\frac{7}{5} = -1.4 Now, order the decimal values from least to greatest. The only positive number, 1.25, is the greatest. Comparing the negative numbers, -1.5 is less than -1.4. So, the order is −1.5<−1.4<1.25-1.5 < -1.4 < 1.25. This corresponds to the order a<c<ba < c < b.

Question 17

Let A=56−25A = \frac{5}{6} - \frac{2}{5} and B=23−14B = \frac{2}{3} - \frac{1}{4}. Which of the following statements is true?

  1. A<BA < B
  2. A>BA > B (correct answer)
  3. A=BA = B
  4. A+B=1A + B = 1
Explanation: First, calculate the value of A: A=56−25A = \frac{5}{6} - \frac{2}{5}. The common denominator is 30. A=2530−1230=1330A = \frac{25}{30} - \frac{12}{30} = \frac{13}{30}. Next, calculate the value of B: B=23−14B = \frac{2}{3} - \frac{1}{4}. The common denominator is 12. B=812−312=512B = \frac{8}{12} - \frac{3}{12} = \frac{5}{12}. Now, compare A and B. To compare 1330\frac{13}{30} and 512\frac{5}{12}, find a common denominator, which is 60. A=1330=2660A = \frac{13}{30} = \frac{26}{60} and B=512=2560B = \frac{5}{12} = \frac{25}{60}. Since 2660>2560\frac{26}{60} > \frac{25}{60}, it follows that A>BA > B.

Question 18

Which of the following inequalities is a true statement?

  1. −58>−35-\frac{5}{8} > -\frac{3}{5}
  2. −79<−67-\frac{7}{9} < -\frac{6}{7}
  3. −23>−58-\frac{2}{3} > -\frac{5}{8}
  4. −34<−811-\frac{3}{4} < -\frac{8}{11} (correct answer)
Explanation: To compare the negative fractions, we can convert them to decimals or use a common denominator. For negative numbers, the number with the smaller absolute value is greater. A) −58=−0.625-\frac{5}{8} = -0.625 and −35=−0.6-\frac{3}{5} = -0.6. Since −0.625<−0.6-0.625 < -0.6, this statement is false. B) −79≈−0.778-\frac{7}{9} \approx -0.778 and −67≈−0.857-\frac{6}{7} \approx -0.857. Since −0.778>−0.857-0.778 > -0.857, this statement is false. C) −23≈−0.667-\frac{2}{3} \approx -0.667 and −58=−0.625-\frac{5}{8} = -0.625. Since −0.667<−0.625-0.667 < -0.625, this statement is false. D) −34=−0.75-\frac{3}{4} = -0.75 and −811≈−0.727-\frac{8}{11} \approx -0.727. Since −0.75<−0.727-0.75 < -0.727, this statement is true.

Question 19

Which of the following rational numbers is located between 47\frac{4}{7} and 58\frac{5}{8} on a number line?

  1. 916\frac{9}{16}
  2. 35\frac{3}{5} (correct answer)
  3. 57\frac{5}{7}
  4. 78\frac{7}{8}
Explanation: To find a number between 47\frac{4}{7} and 58\frac{5}{8}, we can convert the fractions to decimals. 47≈0.5714\frac{4}{7} \approx 0.5714 and 58=0.625\frac{5}{8} = 0.625. We need to find the answer choice that falls between these two values. A) 916=0.5625\frac{9}{16} = 0.5625, which is less than 0.5714. B) 35=0.6\frac{3}{5} = 0.6, which is between 0.5714 and 0.625. C) 57≈0.714\frac{5}{7} \approx 0.714, which is greater than 0.625. D) 78=0.875\frac{7}{8} = 0.875, which is greater than 0.625.

Question 20

A student claims that −712>−35\frac{-7}{12} > -\frac{3}{5} because "7 is greater than 3 and 12 is greater than 5." What is the correct relationship between these fractions?

  1. The student is correct; −712>−35\frac{-7}{12} > -\frac{3}{5} because 7>37 > 3 and 12>512 > 5
  2. The student is incorrect; −712<−35\frac{-7}{12} < -\frac{3}{5} because both fractions are negative
  3. The student is incorrect; −712>−35\frac{-7}{12} > -\frac{3}{5} but not for the stated reason (correct answer)
  4. The student is incorrect; −712=−35\frac{-7}{12} = -\frac{3}{5} when converted to decimals
Explanation: To compare −712\frac{-7}{12} and −35-\frac{3}{5}, find a common denominator. LCM of 12 and 5 is 60. −712=−3560\frac{-7}{12} = \frac{-35}{60} and −35=−3660-\frac{3}{5} = \frac{-36}{60}. Since −35>−36-35 > -36, we have −712>−35\frac{-7}{12} > -\frac{3}{5}. The student's conclusion is correct, but the reasoning is flawed. You cannot compare negative fractions by simply comparing the absolute values of numerators and denominators. The correct comparison requires finding equivalent fractions with the same denominator or converting to decimals: −712≈−0.583\frac{-7}{12} ≈ -0.583 and −35=−0.6-\frac{3}{5} = -0.6. Since −0.583>−0.6-0.583 > -0.6, the inequality −712>−35\frac{-7}{12} > -\frac{3}{5} is true.