Praxis Math Quiz: Perform Rational Operations
20 questions · exam conditions
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Perform Rational OperationsQuestion 1 of 20

A container holds 20 gallons of water. One-fourth of the water is used for gardening. Then, two-fifths of the remaining water is used for cleaning. How many gallons of water are left in the container?

7 gallons
9 gallons
11 gallons
13 gallons
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Praxis Math Quiz

Praxis Math Quiz: Perform Rational Operations

Practice Perform Rational Operations in Praxis Math 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 Perform Rational Operations, giving you a quick way to practice the rules, question types, and explanations that matter most for Praxis Math.

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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

A container holds 20 gallons of water. One-fourth of the water is used for gardening. Then, two-fifths of the remaining water is used for cleaning. How many gallons of water are left in the container?

  1. 7 gallons
  2. 9 gallons (correct answer)
  3. 11 gallons
  4. 13 gallons
Explanation: First, calculate the amount of water used for gardening: 14×20=5\frac{1}{4} \times 20 = 5 gallons. The remaining water is 20−5=1520 - 5 = 15 gallons. Next, calculate the amount used for cleaning from the remainder: 25×15=6\frac{2}{5} \times 15 = 6 gallons. The final amount of water left is 15−6=915 - 6 = 9 gallons.

Question 2

Which of the following is the value of the expression (−1.5)2−(0.5÷−0.2)(-1.5)^2 - (0.5 \div -0.2)?

  1. -4.75
  2. -0.25
  3. 0.25
  4. 4.75 (correct answer)
Explanation: Following the order of operations (PEMDAS/BODMAS): First, evaluate the exponent: (−1.5)2=2.25(-1.5)^2 = 2.25. Next, evaluate the division inside the parentheses: 0.5÷−0.2=−2.50.5 \div -0.2 = -2.5. Finally, perform the subtraction: 2.25−(−2.5)=2.25+2.5=4.752.25 - (-2.5) = 2.25 + 2.5 = 4.75.

Question 3

A carpenter has a board that is 8138\frac{1}{3} feet long. He needs to cut as many pieces as possible that are each 23\frac{2}{3} feet long. After he cuts the maximum number of full pieces, what is the length, in feet, of the remaining board?

  1. 13\frac{1}{3} (correct answer)
  2. 12\frac{1}{2}
  3. 23\frac{2}{3}
  4. 0
Explanation: First, find how many 23\frac{2}{3}-foot pieces can be cut from the 8138\frac{1}{3}-foot board. This is a division problem: 813÷23=253÷23=253×32=252=12.58\frac{1}{3} \div \frac{2}{3} = \frac{25}{3} \div \frac{2}{3} = \frac{25}{3} \times \frac{3}{2} = \frac{25}{2} = 12.5. The carpenter can cut 12 full pieces. The total length of these 12 pieces is 12×23=243=812 \times \frac{2}{3} = \frac{24}{3} = 8 feet. The length of the remaining board is the original length minus the length of the cut pieces: 813−8=138\frac{1}{3} - 8 = \frac{1}{3} foot.

Question 4

Over a three-day period, the daily change in a pond's water level was recorded as an increase of 2.5 inches, a decrease of 4.1 inches, and an increase of 0.7 inches. What was the average daily change in the water level for this period?

  1. -0.9 inches
  2. -0.3 inches (correct answer)
  3. 0.3 inches
  4. 2.43 inches
Explanation: To find the average daily change, first sum the changes: 2.5+(−4.1)+0.7=−1.6+0.7=−0.92.5 + (-4.1) + 0.7 = -1.6 + 0.7 = -0.9 inches. Then, divide the total change by the number of days, which is 3. The average change is −0.9÷3=−0.3-0.9 \div 3 = -0.3 inches.

Question 5

A rectangular garden has a length of ( 5125\frac{1}{2} ) feet and a width of ( 3123\frac{1}{2} ) feet. A path that is ( 14\frac{1}{4} ) foot wide is built around the outside of the garden. What is the perimeter, in feet, of the outer edge of the path?

  1. 18
  2. 19
  3. 20 (correct answer)
  4. 22
Explanation: The path adds (14)( \frac{1}{4} ) foot to both sides of the length and both sides of the width. The new length is ( 5\frac{1}{2} + 2 \times \frac{1}{4} \= 5\frac{1}{2} + \frac{1}{2} \= 6 ) feet. The new width is ( 3\frac{1}{2} + 2 \times \frac{1}{4} \= 3\frac{1}{2} + \frac{1}{2} \= 4 ) feet. The perimeter of the outer edge of the path is calculated as ( 2 \times (\text{new length} + \text{new width}) \= 2 \times (6 + 4) \= 2 \times 10 \= 20 ) feet.

Question 6

A car's gas tank was 78\frac{7}{8} full. The driver used 12\frac{1}{2} of the gas in the tank to drive to a meeting. Then, they added 5 gallons of gas to the tank, which was found to be 34\frac{3}{4} full. What is the total capacity, in gallons, of the gas tank?

  1. 12
  2. 14
  3. 16 (correct answer)
  4. 20
Explanation: Let C be the total capacity of the tank. Initially, the tank had 78C\frac{7}{8}C gallons. The driver used half of this amount, so the gas used was 12×78C=716C\frac{1}{2} \times \frac{7}{8}C = \frac{7}{16}C. The amount of gas remaining after the drive was 78C−716C=1416C−716C=716C\frac{7}{8}C - \frac{7}{16}C = \frac{14}{16}C - \frac{7}{16}C = \frac{7}{16}C. After adding 5 gallons, the tank was 34\frac{3}{4} full. So, 716C+5=34C\frac{7}{16}C + 5 = \frac{3}{4}C. To solve for C, subtract 716C\frac{7}{16}C from both sides: 5=34C−716C=1216C−716C=516C5 = \frac{3}{4}C - \frac{7}{16}C = \frac{12}{16}C - \frac{7}{16}C = \frac{5}{16}C. Now, solve for C: C=5÷516=5×165=16C = 5 \div \frac{5}{16} = 5 \times \frac{16}{5} = 16. The total capacity is 16 gallons.

Question 7

A worker is filling a pool. The water level starts at 1141\frac{1}{4} feet. For 3 hours, the water level rises at a rate of 12\frac{1}{2} foot per hour. Then, a small leak causes the water level to drop by a total of 13\frac{1}{3} foot. What is the final water level in feet?

  1. 2142\frac{1}{4}
  2. 25122\frac{5}{12} (correct answer)
  3. 27122\frac{7}{12}
  4. 2342\frac{3}{4}
Explanation: First, calculate the total rise in water level over 3 hours: 3×12=323 \times \frac{1}{2} = \frac{3}{2} feet. Add this to the starting level: 114+32=54+64=1141\frac{1}{4} + \frac{3}{2} = \frac{5}{4} + \frac{6}{4} = \frac{11}{4} feet. Now, subtract the amount the water level dropped due to the leak: 114−13\frac{11}{4} - \frac{1}{3}. Find a common denominator, which is 12: 3312−412=2912\frac{33}{12} - \frac{4}{12} = \frac{29}{12}. Converting this improper fraction to a mixed number gives 25122\frac{5}{12} feet.

Question 8

An account has a balance of $150.40. A deposit of $49.60 is made. Then, two checks are written: one for $30 and another for 15\frac{1}{5} of the balance that was in the account after the deposit. What is the final balance in the account?

  1. $130.00 (correct answer)
  2. $130.80
  3. $140.00
  4. $160.00
Explanation: First, calculate the balance after the deposit: \150.40 + $49.60 = $200.00.Next,calculatetheamountofthesecondcheck,whichis. Next, calculate the amount of the second check, which is \frac{1}{5}ofthisnewbalance:of this new balance:\frac{1}{5} \times $200.00 = $40.00.Now,subtracttheamountsofbothchecksfromthebalance:. Now, subtract the amounts of both checks from the balance: $200.00 - $30.00 - $40.00 = $130.00$.

Question 9

Three friends are sharing the cost of a gift that costs $72. The first friend pays 13\frac{1}{3} of the cost. The second friend pays 58\frac{5}{8} of the remaining cost. The third friend pays the rest. How much more did the second friend pay than the third friend?

  1. $6
  2. $12 (correct answer)
  3. $18
  4. $30
Explanation: First, calculate the first friend's payment: \frac{1}{3} \times \72 = $24.Next,calculatetheremainingcost:. Next, calculate the remaining cost: $72 - $24 = $48.Then,calculatethesecondfriend′spayment:. Then, calculate the second friend's payment: \frac{5}{8} \times $48 = 5 \times $6 = $30.Finally,calculatethethirdfriend′spayment,whichistherestoftheremainingcost:. Finally, calculate the third friend's payment, which is the rest of the remaining cost: $48 - $30 = $18.Thequestionaskshowmuchmorethesecondfriendpaidthanthethirdfriend:. The question asks how much more the second friend paid than the third friend: $30 - $18 = $12$.

Question 10

What is the value of the expression (12+13)÷(12−13)(\frac{1}{2} + \frac{1}{3}) \div (\frac{1}{2} - \frac{1}{3})?

  1. 16\frac{1}{6}
  2. 1
  3. 5 (correct answer)
  4. 6
Explanation: First, evaluate the expression in the first set of parentheses: 12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}. Next, evaluate the expression in the second set of parentheses: 12−13=36−26=16\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}. Finally, perform the division: 56÷16=56×61=5\frac{5}{6} \div \frac{1}{6} = \frac{5}{6} \times \frac{6}{1} = 5.

Question 11

A hiker starts at an elevation of 240.5 feet above sea level. He hikes up a trail, increasing his elevation by 51234512\frac{3}{4} feet. He then takes a different trail down, decreasing his elevation by 625.25 feet. What is his final elevation in feet relative to sea level?

  1. 128.00 (correct answer)
  2. 128.50
  3. 138.00
  4. 138.50
Explanation: First, convert all numbers to decimals. The starting elevation is 240.5 feet. The increase is 51234=512.75512\frac{3}{4} = 512.75 feet. The decrease is 625.25 feet. The final elevation is calculated by: 240.5+512.75−625.25240.5 + 512.75 - 625.25. First add: 240.5+512.75=753.25240.5 + 512.75 = 753.25. Then subtract: 753.25−625.25=128.00753.25 - 625.25 = 128.00. The final elevation is 128.00 feet above sea level.

Question 12

From a piece of cloth measuring 20 yards, a tailor cuts a piece that is 5135\frac{1}{3} yards long and another piece that is 7127\frac{1}{2} yards long. The remaining piece is then divided into two equal parts. What is the length of each part?

  1. 37123\frac{7}{12} yards (correct answer)
  2. 3123\frac{1}{2} yards
  3. 7167\frac{1}{6} yards
  4. 125612\frac{5}{6} yards
Explanation: First, find the total length of the two pieces cut from the cloth: 513+712=526+736=12565\frac{1}{3} + 7\frac{1}{2} = 5\frac{2}{6} + 7\frac{3}{6} = 12\frac{5}{6} yards. Next, find the length of the remaining piece: 20−1256=1966−1256=71620 - 12\frac{5}{6} = 19\frac{6}{6} - 12\frac{5}{6} = 7\frac{1}{6} yards. Finally, divide the remaining piece into two equal parts: 716÷2=436÷2=436×12=43127\frac{1}{6} \div 2 = \frac{43}{6} \div 2 = \frac{43}{6} \times \frac{1}{2} = \frac{43}{12}. Convert this to a mixed number: 37123\frac{7}{12} yards.

Question 13

Kalinda has a budget of $150 for school supplies. She spends 15\frac{1}{5} of her budget on notebooks and 12\frac{1}{2} of the remaining budget on a calculator. How much money does she have left?

  1. $30
  2. $45
  3. $60 (correct answer)
  4. $75
Explanation: First, calculate the amount spent on notebooks: \frac{1}{5} \times \150 = $30.Next,calculatetheremainingbudget:. Next, calculate the remaining budget: $150 - $30 = $120.Then,calculatetheamountspentonthecalculator,whichishalfoftheremainingbudget:. Then, calculate the amount spent on the calculator, which is half of the remaining budget: \frac{1}{2} \times $120 = $60.Finally,calculatethemoneyleftafterbuyingthecalculator:. Finally, calculate the money left after buying the calculator: $120 - $60 = $60$.

Question 14

The temperature at 6:00 a.m. was −8.4∘F-8.4^{\circ}F. By noon, the temperature had risen by 15.7∘F15.7^{\circ}F. By 6:00 p.m., the temperature had dropped by 12\frac{1}{2} of the temperature at noon. What was the temperature at 6:00 p.m.?

  1. −1.0∘F-1.0^{\circ}F
  2. 3.65∘F3.65^{\circ}F (correct answer)
  3. 7.3∘F7.3^{\circ}F
  4. 11.55∘F11.55^{\circ}F
Explanation: First, find the temperature at noon: −8.4+15.7=7.3∘F-8.4 + 15.7 = 7.3^{\circ}F. Next, find the temperature drop by 6:00 p.m. The drop is half of the noon temperature: 12×7.3=3.65∘F\frac{1}{2} \times 7.3 = 3.65^{\circ}F. The temperature at 6:00 p.m. is the noon temperature minus the drop: 7.3−3.65=3.65∘F7.3 - 3.65 = 3.65^{\circ}F.

Question 15

Jamal is located on a number line at the point -2.5. He moves 4144\frac{1}{4} units in the positive direction, and then moves 6.5 units in the negative direction. What is his final position on the number line?

  1. -4.75 (correct answer)
  2. -3.75
  3. -0.75
  4. 1.75
Explanation: Start at -2.5. Moving 4144\frac{1}{4} units in the positive direction is adding 4.25. The new position is −2.5+4.25=1.75-2.5 + 4.25 = 1.75. From this point, moving 6.5 units in the negative direction is subtracting 6.5. The final position is 1.75−6.5=−4.751.75 - 6.5 = -4.75.

Question 16

A full bucket of water weighs 261226\frac{1}{2} kilograms. The empty bucket weighs 2122\frac{1}{2} kilograms. If 13\frac{1}{3} of the water is removed from the full bucket, what is the total weight of the bucket and the remaining water?

  1. 161216\frac{1}{2} kilograms
  2. 172317\frac{2}{3} kilograms
  3. 181218\frac{1}{2} kilograms (correct answer)
  4. 2424 kilograms
Explanation: First, find the weight of the water alone by subtracting the bucket's weight from the total weight: 2612−212=2426\frac{1}{2} - 2\frac{1}{2} = 24 kilograms. Next, calculate the amount of water remaining. If 13\frac{1}{3} is removed, then 1−13=231 - \frac{1}{3} = \frac{2}{3} of the water remains. The weight of the remaining water is 23×24=16\frac{2}{3} \times 24 = 16 kilograms. Finally, find the total weight by adding the weight of the bucket to the weight of the remaining water: 16+212=181216 + 2\frac{1}{2} = 18\frac{1}{2} kilograms.

Question 17

A stock is priced at $50.00. On Monday, its value increases by 15\frac{1}{5}. On Tuesday, its value decreases by 110\frac{1}{10} of its value at the end of Monday. What is the final price of the stock?

  1. $59.00
  2. $55.00
  3. $54.00 (correct answer)
  4. $44.00
Explanation: First, calculate the increase on Monday: \frac{1}{5} \times \50.00 = $10.00.ThepriceattheendofMondayis. The price at the end of Monday is $50.00 + $10.00 = $60.00.Next,calculatethedecreaseonTuesdaybasedonthenewprice:. Next, calculate the decrease on Tuesday based on the new price: \frac{1}{10} \times $60.00 = $6.00.Thefinalpriceis. The final price is $60.00 - $6.00 = $54.00$.

Question 18

A machine can produce a batch of widgets in 4124\frac{1}{2} hours. A new machine can do the same job in 23\frac{2}{3} of the time. If the new machine starts a batch at 8:00 a.m., at what time will it finish?

  1. 10:30 a.m.
  2. 11:00 a.m. (correct answer)
  3. 11:30 a.m.
  4. 12:00 p.m.
Explanation: First, calculate the time it takes the new machine. This is 23\frac{2}{3} of 4124\frac{1}{2} hours. Convert the mixed number to an improper fraction: 412=924\frac{1}{2} = \frac{9}{2}. Now multiply: 23×92=186=3\frac{2}{3} \times \frac{9}{2} = \frac{18}{6} = 3 hours. If the machine starts at 8:00 a.m. and runs for 3 hours, it will finish at 11:00 a.m.

Question 19

A piece of wood is 10 feet long. A carpenter cuts off a piece that is 3583\frac{5}{8} feet long. He then cuts the remaining piece into 3 equal lengths. What is the length, in feet, of each of these 3 equal pieces?

  1. 2182\frac{1}{8} (correct answer)
  2. 2382\frac{3}{8}
  3. 33243\frac{3}{24}
  4. 6386\frac{3}{8}
Explanation: First, find the length of the remaining piece of wood after the first cut. This requires subtracting 3583\frac{5}{8} from 10. 10−358=988−358=63810 - 3\frac{5}{8} = 9\frac{8}{8} - 3\frac{5}{8} = 6\frac{3}{8} feet. Next, divide the remaining piece into 3 equal lengths: 638÷36\frac{3}{8} \div 3. Convert the mixed number to an improper fraction: 518÷3=518×13=5124\frac{51}{8} \div 3 = \frac{51}{8} \times \frac{1}{3} = \frac{51}{24}. Simplify the fraction: 178\frac{17}{8}. Convert back to a mixed number: 2182\frac{1}{8} feet.

Question 20

Which of the following expressions has the greatest value?

  1. −45÷−23-\frac{4}{5} \div -\frac{2}{3} (correct answer)
  2. −12+43-\frac{1}{2} + \frac{4}{3}
  3. 34−(−13)\frac{3}{4} - (-\frac{1}{3})
  4. (−32)×(−23)( -\frac{3}{2} ) \times ( -\frac{2}{3} )
Explanation: To find the expression with the greatest value, we must evaluate each one. A) −45÷−23=−45×−32=1210=1.2-\frac{4}{5} \div -\frac{2}{3} = -\frac{4}{5} \times -\frac{3}{2} = \frac{12}{10} = 1.2. B) −12+43=−36+86=56≈0.833-\frac{1}{2} + \frac{4}{3} = -\frac{3}{6} + \frac{8}{6} = \frac{5}{6} \approx 0.833. C) 34−(−13)=34+13=912+412=1312≈1.083\frac{3}{4} - (-\frac{1}{3}) = \frac{3}{4} + \frac{1}{3} = \frac{9}{12} + \frac{4}{12} = \frac{13}{12} \approx 1.083. D) (−32)×(−23)=1( -\frac{3}{2} ) \times ( -\frac{2}{3} ) = 1. Comparing the results: 1.2, 0.833..., 1.083..., and 1. The greatest value is 1.2, which corresponds to option A.