ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Rational Number Operations
20 questions · exam conditions
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Rational Number OperationsQuestion 1 of 20

A stock's price started at $45. On Monday, it lost 110\frac{1}{10} of its value. On Tuesday, the new price gained 15\frac{1}{5} of its value. What was the price of the stock at the end of Tuesday?

$48.00
$48.60
$49.50
$54.00
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ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz

ACCUPLACER Quantitative Reasoning, Algebra & Statistics Quiz: Rational Number Operations

Practice Rational Number Operations 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 Rational Number Operations, 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.

All questions

Question 1

A stock's price started at $45. On Monday, it lost 110\frac{1}{10} of its value. On Tuesday, the new price gained 15\frac{1}{5} of its value. What was the price of the stock at the end of Tuesday?

  1. $48.00
  2. $48.60 (correct answer)
  3. $49.50
  4. $54.00
Explanation: First, calculate the price after Monday's loss. The loss was (\frac{1}{10} \times 45 = 4.50\). The new price was \(45 - 4.50=4.50 = 40.50). Next, calculate Tuesday's gain based on this new price. The gain was (\frac{1}{5} \times 40.50 = 8.10\). The final price is \(40.50 + 8.10=8.10 = 48.60).

Question 2

Sam had $240.00 in his bank account. He spent 18\frac{1}{8} of the money on a concert ticket. He then deposited a check for $67.50. Finally, he withdrew 13\frac{1}{3} of the new balance. How much money is left in his account?

  1. $92.50
  2. $140.00
  3. $185.00 (correct answer)
  4. $197.50
Explanation: First, calculate the cost of the ticket: (\frac{1}{8} \times 240 = 30\). The new balance is \(240 - 30=30 = 210). Next, add the deposit: (210+210 + 67.50 = 277.50\). Then, calculate the withdrawal amount: \(\frac{1}{3} \times 277.50 = 92.50). Finally, find the remaining balance: (277.50−277.50 - 92.50 = $185.00).

Question 3

Which of the following is equivalent to 1−132+12\frac{1 - \frac{1}{3}}{2 + \frac{1}{2}}?

  1. 415\frac{4}{15} (correct answer)
  2. 49\frac{4}{9}
  3. 1
  4. 53\frac{5}{3}
Explanation: First, simplify the numerator: 1−13=33−13=231 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}. Next, simplify the denominator: 2+12=42+12=522 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}. The expression is now a division of two fractions: 2/35/2\frac{2/3}{5/2}. To divide by a fraction, multiply by its reciprocal: 23×25=415\frac{2}{3} \times \frac{2}{5} = \frac{4}{15}.

Question 4

A length of ribbon 101210\frac{1}{2} inches long is cut into smaller pieces, each 34\frac{3}{4} of an inch long. What is the maximum number of full pieces that can be cut, and what is the length of the ribbon left over?

  1. 13 pieces with 12\frac{1}{2} inch left over
  2. 14 pieces with 0 inches left over (correct answer)
  3. 14 pieces with 12\frac{1}{2} inch left over
  4. 15 pieces with 0 inches left over
Explanation: To find how many pieces can be cut, divide the total length by the length of each piece. Convert to improper fractions: 1012=21210\frac{1}{2} = \frac{21}{2}. The division is 212÷34=212×43=846=14\frac{21}{2} \div \frac{3}{4} = \frac{21}{2} \times \frac{4}{3} = \frac{84}{6} = 14. Since the result is an integer, exactly 14 full pieces can be cut with 0 inches left over.

Question 5

A carpenter has a piece of wood that is 121212\frac{1}{2} feet long. She cuts off a piece that is 3343\frac{3}{4} feet long. She then wants to cut the remaining piece into 5 equal-length shelves. What is the length of each shelf in feet?

  1. 1341\frac{3}{4} (correct answer)
  2. 2122\frac{1}{2}
  3. 3143\frac{1}{4}
  4. 8348\frac{3}{4}
Explanation: First, find the length of the remaining piece of wood by subtracting the cut piece from the original length: 1212−334=252−154=504−154=35412\frac{1}{2} - 3\frac{3}{4} = \frac{25}{2} - \frac{15}{4} = \frac{50}{4} - \frac{15}{4} = \frac{35}{4} feet. Next, divide the remaining length by the number of shelves, 5: 354÷5=354×15=3520=74=134\frac{35}{4} \div 5 = \frac{35}{4} \times \frac{1}{5} = \frac{35}{20} = \frac{7}{4} = 1\frac{3}{4} feet.

Question 6

A rectangular garden has a length of 5.5 meters and a width of 3143\frac{1}{4} meters. Fencing costs $8 per meter. What is the total cost to build a fence around the entire garden?

  1. $70.00
  2. $136.50
  3. $140.00 (correct answer)
  4. $143.00
Explanation: First, calculate the perimeter of the garden. The formula for the perimeter of a rectangle is P=2(L+W)P = 2(L+W). Convert the width to a decimal: 314=3.253\frac{1}{4} = 3.25 meters. Now find the perimeter: P=2(5.5+3.25)=2(8.75)=17.5P = 2(5.5 + 3.25) = 2(8.75) = 17.5 meters. Finally, calculate the total cost by multiplying the perimeter by the cost per meter: (17.5 \times 8 = $140.00).

Question 7

Let P=(−2)÷14P = (-2) \div \frac{1}{4} and Q=(−12)×4Q = (-\frac{1}{2}) \times 4. Which of the following statements is true?

  1. P>QP > Q
  2. Q>PQ > P (correct answer)
  3. P=QP = Q
  4. P=−QP = -Q
Explanation: First, calculate the value of P: P=−2÷14=−2×4=−8P = -2 \div \frac{1}{4} = -2 \times 4 = -8. Next, calculate the value of Q: Q=−12×4=−2Q = -\frac{1}{2} \times 4 = -2. Now compare the two values. Since -2 is greater than -8, the true statement is Q>PQ > P.

Question 8

A recipe calls for 23\frac{2}{3} cup of flour for every 1141\frac{1}{4} cups of sugar. If Maria wants to make a batch using 2122\frac{1}{2} cups of sugar, how many cups of flour will she need?

  1. 1131\frac{1}{3} cups (correct answer)
  2. 1251\frac{2}{5} cups
  3. 17151\frac{7}{15} cups
  4. 1121\frac{1}{2} cups
Explanation: First, set up the proportion: 2/3114=x212\frac{2/3}{1\frac{1}{4}} = \frac{x}{2\frac{1}{2}}. Convert mixed numbers to improper fractions: 2/35/4=x5/2\frac{2/3}{5/4} = \frac{x}{5/2}. Cross multiply: 23×52=x×54\frac{2}{3} \times \frac{5}{2} = x \times \frac{5}{4}. This gives 106=5x4\frac{10}{6} = \frac{5x}{4}. Solving for x: x=106×45=4030=43=113x = \frac{10}{6} \times \frac{4}{5} = \frac{40}{30} = \frac{4}{3} = 1\frac{1}{3}. Choice B results from incorrectly multiplying 23×212\frac{2}{3} \times 2\frac{1}{2} directly. Choice C comes from calculation errors in cross multiplication. Choice D results from rounding errors or mishandling the fraction arithmetic.

Question 9

A carpenter cuts a board that is 4384\frac{3}{8} feet long into three pieces. The first piece is 1561\frac{5}{6} feet long, and the second piece is 0.750.75 feet long. What is the length of the third piece in feet?

  1. 119241\frac{19}{24} feet (correct answer)
  2. 1581\frac{5}{8} feet
  3. 111121\frac{11}{12} feet
  4. 113161\frac{13}{16} feet
Explanation: The third piece equals the total length minus the sum of the first two pieces. Convert all measurements to the same form: 438=3584\frac{3}{8} = \frac{35}{8}, 156=1161\frac{5}{6} = \frac{11}{6}, and 0.75=340.75 = \frac{3}{4}. Find the sum of the first two pieces using LCD 12: 116=2212\frac{11}{6} = \frac{22}{12} and 34=912\frac{3}{4} = \frac{9}{12}, so 2212+912=3112\frac{22}{12} + \frac{9}{12} = \frac{31}{12}. Now subtract from the total using LCD 24: 358=10524\frac{35}{8} = \frac{105}{24} and 3112=6224\frac{31}{12} = \frac{62}{24}. Therefore, 10524−6224=4324=11924\frac{105}{24} - \frac{62}{24} = \frac{43}{24} = 1\frac{19}{24}. Choice B results from converting 0.75 incorrectly to 58\frac{5}{8}. Choice C comes from calculation errors when finding the LCD. Choice D results from mishandling the mixed number conversion.

Question 10

A recipe for fruit punch calls for mixing cranberry juice and apple juice in the ratio 2:32:3. If the total amount of punch needed is 3343\frac{3}{4} quarts, and cranberry juice costs $2.40 per quart while apple juice costs $1.80 per quart, what is the total cost of the juice needed?

  1. $7.65 (correct answer)
  2. $8.10
  3. $7.85
  4. $8.25
Explanation: The ratio 2:32:3 means cranberry juice makes up 25\frac{2}{5} of the total and apple juice makes up 35\frac{3}{5}. Total punch needed: 334=1543\frac{3}{4} = \frac{15}{4} quarts. Cranberry juice needed: 25×154=3020=32=1.5\frac{2}{5} \times \frac{15}{4} = \frac{30}{20} = \frac{3}{2} = 1.5 quarts. Apple juice needed: 35×154=4520=94=2.25\frac{3}{5} \times \frac{15}{4} = \frac{45}{20} = \frac{9}{4} = 2.25 quarts. Cost calculation: Cranberry juice cost = 1.5×$2.40=$3.601.5 \times \$2.40 = \$3.60. Apple juice cost = 2.25×$1.80=$4.052.25 \times \$1.80 = \$4.05. Total cost = $3.60+$4.05=$7.65\$3.60 + \$4.05 = \$7.65. Choice B results from incorrectly calculating the ratio as 23\frac{2}{3} and 13\frac{1}{3} instead of 25\frac{2}{5} and 35\frac{3}{5}. Choice C comes from arithmetic errors in the multiplication. Choice D results from using the wrong total amount or swapping the juice prices.

Question 11

The product of two numbers is −0.75-0.75. If one of the numbers is 4124\frac{1}{2}, what is the other number?

  1. −16-\frac{1}{6} (correct answer)
  2. −13-\frac{1}{3}
  3. 16\frac{1}{6}
  4. 13\frac{1}{3}
Explanation: Let the unknown number be xx. The equation is 412×x=−0.754\frac{1}{2} \times x = -0.75. Convert both numbers to fractions to make the division easier. 412=924\frac{1}{2} = \frac{9}{2} and −0.75=−34-0.75 = -\frac{3}{4}. The equation is 92x=−34\frac{9}{2}x = -\frac{3}{4}. To solve for xx, divide −34-\frac{3}{4} by 92\frac{9}{2}, which is the same as multiplying by its reciprocal: x=−34×29=−636x = -\frac{3}{4} \times \frac{2}{9} = -\frac{6}{36}. Simplifying this fraction gives −16-\frac{1}{6}.

Question 12

A recipe for cookies requires 1341\frac{3}{4} cups of flour and makes 2 dozen cookies. A baker wants to make 5 dozen cookies. How many cups of flour are needed?

  1. 2382\frac{3}{8}
  2. 4144\frac{1}{4}
  3. 4384\frac{3}{8} (correct answer)
  4. 5145\frac{1}{4}
Explanation: First, find the scaling factor for the recipe by dividing the desired amount by the original amount: 5÷2=2.55 \div 2 = 2.5 or 52\frac{5}{2}. Next, multiply the amount of flour by this scaling factor. Convert 1341\frac{3}{4} to an improper fraction, which is 74\frac{7}{4}. Now multiply: 74×52=358\frac{7}{4} \times \frac{5}{2} = \frac{35}{8}. Converting back to a mixed number gives 4384\frac{3}{8} cups.

Question 13

What is the value of the expression 13−(−25)÷(−45)\frac{1}{3} - \left(-\frac{2}{5}\right) \div \left(-\frac{4}{5}\right)?

  1. −1112-\frac{11}{12}
  2. −16-\frac{1}{6} (correct answer)
  3. 16\frac{1}{6}
  4. 56\frac{5}{6}
Explanation: According to the order of operations, perform the division first: (−25)÷(−45)=(−25)×(−54)=1020=12\left(-\frac{2}{5}\right) \div \left(-\frac{4}{5}\right) = \left(-\frac{2}{5}\right) \times \left(-\frac{5}{4}\right) = \frac{10}{20} = \frac{1}{2}. Now, substitute this result back into the expression: 13−12\frac{1}{3} - \frac{1}{2}. Find a common denominator to subtract: 26−36=−16\frac{2}{6} - \frac{3}{6} = -\frac{1}{6}.

Question 14

A number nn is decreased by 15\frac{1}{5} of its value. The result is then multiplied by −2.5-2.5. If the final result is 12, what was the original number nn?

  1. -24
  2. -6 (correct answer)
  3. -4
  4. 6
Explanation: Work backwards from the final result. The result after multiplication is 12. Before being multiplied by -2.5, the number was 12÷(−2.5)=−4.812 \div (-2.5) = -4.8. This number, -4.8, is the result of decreasing nn by 15\frac{1}{5} of its value. This means n−15n=−4.8n - \frac{1}{5}n = -4.8, which simplifies to 45n=−4.8\frac{4}{5}n = -4.8. To find nn, divide -4.8 by 45\frac{4}{5}: n=−4.8÷45=−4.8×54=−1.2×5=−6n = -4.8 \div \frac{4}{5} = -4.8 \times \frac{5}{4} = -1.2 \times 5 = -6.

Question 15

What is the average (arithmetic mean) of the following three numbers: -4, 72\frac{7}{2}, and 5.5?

  1. 1231\frac{2}{3} (correct answer)
  2. 2.52.5
  3. 4134\frac{1}{3}
  4. 5
Explanation: To find the average, first sum the numbers. It is easiest to convert all numbers to a common format, like decimals: −4-4, 3.53.5, and 5.55.5. The sum is −4+3.5+5.5=−4+9=5-4 + 3.5 + 5.5 = -4 + 9 = 5. Next, divide the sum by the count of the numbers, which is 3. The average is 5÷3=535 \div 3 = \frac{5}{3}. As a mixed number, this is 1231\frac{2}{3}.

Question 16

A tank is 34\frac{3}{4} full of water. After 15.5 gallons are used, the tank is 13\frac{1}{3} full. What is the total capacity of the tank in gallons?

  1. 30.6
  2. 37.2 (correct answer)
  3. 42.5
  4. 50.0
Explanation: Let CC be the total capacity of the tank. The change in the amount of water is the fraction of the tank that was drained: 34−13=912−412=512\frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}. This fraction of the total capacity is equal to the 15.5 gallons that were used. So, 512C=15.5\frac{5}{12}C = 15.5. To find CC, multiply 15.5 by the reciprocal of 512\frac{5}{12}: C=15.5×125=3.1×12=37.2C = 15.5 \times \frac{12}{5} = 3.1 \times 12 = 37.2 gallons.

Question 17

What is the solution to the equation 23y−16=32\frac{2}{3}y - \frac{1}{6} = \frac{3}{2}?

  1. 34\frac{3}{4}
  2. 109\frac{10}{9}
  3. 2
  4. 52\frac{5}{2} (correct answer)
Explanation: To solve for yy, first isolate the term with yy by adding 16\frac{1}{6} to both sides: 23y=32+16\frac{2}{3}y = \frac{3}{2} + \frac{1}{6}. Find a common denominator for the right side (6): 23y=96+16=106\frac{2}{3}y = \frac{9}{6} + \frac{1}{6} = \frac{10}{6}, which simplifies to 53\frac{5}{3}. The equation is now 23y=53\frac{2}{3}y = \frac{5}{3}. To solve for yy, multiply both sides by the reciprocal of 23\frac{2}{3}, which is 32\frac{3}{2}: y=53×32=156=52y = \frac{5}{3} \times \frac{3}{2} = \frac{15}{6} = \frac{5}{2}.

Question 18

If x=−12x = -\frac{1}{2} and y=23y = \frac{2}{3}, what is the value of the expression 4x−6y4x - 6y?

  1. -6 (correct answer)
  2. -2
  3. 2
  4. 6
Explanation: Substitute the given values for xx and yy into the expression. First, calculate 4x4x: 4(−12)=−24(-\frac{1}{2}) = -2. Next, calculate 6y6y: 6(23)=123=46(\frac{2}{3}) = \frac{12}{3} = 4. Finally, substitute these values back into the expression 4x−6y4x - 6y: −2−4=−6-2 - 4 = -6.

Question 19

The temperature at noon was 5.4∘5.4^\circC. By 6:00 PM, it had dropped by 8128\frac{1}{2} degrees. By midnight, it had dropped another 2.3∘2.3^\circC. What was the temperature at midnight?

  1. −6.2∘-6.2^\circC
  2. −3.1∘-3.1^\circC
  3. −4.8∘-4.8^\circC
  4. −5.4∘-5.4^\circC (correct answer)
Explanation: Start with the initial temperature and subtract the two drops. It's easiest to convert the fraction to a decimal: 812=8.58\frac{1}{2} = 8.5. The calculation is 5.4−8.5−2.35.4 - 8.5 - 2.3. First, 5.4−8.5=−3.15.4 - 8.5 = -3.1. Then, −3.1−2.3=−5.4-3.1 - 2.3 = -5.4. The temperature at midnight was −5.4∘-5.4^\circC.

Question 20

Evaluate the expression (−32)2−(12−3)(-\frac{3}{2})^2 - (\frac{1}{2} - 3).

  1. −14-\frac{1}{4}
  2. 12\frac{1}{2}
  3. 194\frac{19}{4} (correct answer)
  4. 294\frac{29}{4}
Explanation: Following the order of operations, evaluate the exponents and parentheses first. (−32)2=(−32)(−32)=94(-\frac{3}{2})^2 = (-\frac{3}{2})(-\frac{3}{2}) = \frac{9}{4}. Inside the second parenthesis: 12−3=12−62=−52\frac{1}{2} - 3 = \frac{1}{2} - \frac{6}{2} = -\frac{5}{2}. Now the expression is 94−(−52)\frac{9}{4} - (-\frac{5}{2}). Subtracting a negative is the same as adding a positive: 94+52\frac{9}{4} + \frac{5}{2}. Find a common denominator: 94+104=194\frac{9}{4} + \frac{10}{4} = \frac{19}{4}.