HSPT Math Quiz: Perform Fraction Operations
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Perform Fraction OperationsQuestion 1 of 20

A canister of oil was 78\frac{7}{8} full. A cook used 13\frac{1}{3} of the oil in the canister, then added 0.5 liters of oil. If the canister is now 34\frac{3}{4} full, what is the total capacity of the canister in liters?

2.4 L
3.0 L
4.0 L
6.0 L
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HSPT Math Quiz

HSPT Math Quiz: Perform Fraction Operations

Practice Perform Fraction Operations in HSPT Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Perform Fraction Operations, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT 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 canister of oil was 78\frac{7}{8} full. A cook used 13\frac{1}{3} of the oil in the canister, then added 0.5 liters of oil. If the canister is now 34\frac{3}{4} full, what is the total capacity of the canister in liters?

  1. 2.4 L
  2. 3.0 L (correct answer)
  3. 4.0 L
  4. 6.0 L
Explanation: Let C be the total capacity of the canister.
Initial amount of oil: 78C\frac{7}{8}C.
Amount of oil used: 13\frac{1}{3} of 78C\frac{7}{8}C, which is 13×78C=724C\frac{1}{3} \times \frac{7}{8}C = \frac{7}{24}C.
Amount remaining after use: 78C−724C=2124C−724C=1424C=712C\frac{7}{8}C - \frac{7}{24}C = \frac{21}{24}C - \frac{7}{24}C = \frac{14}{24}C = \frac{7}{12}C.
After adding 0.5 liters, the amount is 712C+0.5\frac{7}{12}C + 0.5.
This new amount is equal to 34C\frac{3}{4}C.
Set up the equation: 712C+0.5=34C\frac{7}{12}C + 0.5 = \frac{3}{4}C.
To solve for C, first isolate the terms with C: 0.5=34C−712C0.5 = \frac{3}{4}C - \frac{7}{12}C.
Find a common denominator: 0.5=912C−712C=212C=16C0.5 = \frac{9}{12}C - \frac{7}{12}C = \frac{2}{12}C = \frac{1}{6}C.
So, 12=16C\frac{1}{2} = \frac{1}{6}C. Multiply both sides by 6 to find C: C=6×12=3C = 6 \times \frac{1}{2} = 3.
The total capacity is 3.0 liters.

Question 2

A solution contains 40%40\% acid. How many liters of pure water must be added to 3133\frac{1}{3} liters of this solution to create a mixture that is 25%25\% acid?

  1. 1231\frac{2}{3} liters
  2. 22 liters (correct answer)
  3. 2132\frac{1}{3} liters
  4. 2232\frac{2}{3} liters
Explanation: This is a mixture problem involving concentration percentages. When you see questions about diluting solutions, focus on tracking the amount of pure substance (acid) before and after mixing. Start by finding how much pure acid is in the original solution. You have 313=1033\frac{1}{3} = \frac{10}{3} liters at 40% concentration, so the pure acid amount is 103×0.40=43\frac{10}{3} \times 0.40 = \frac{4}{3} liters of acid. When you add pure water, the amount of acid stays the same, but the total volume increases. If you add xx liters of water, your new total volume becomes 103+x\frac{10}{3} + x liters, still containing 43\frac{4}{3} liters of acid. Set up the equation for 25% concentration: 43103+x=0.25\frac{\frac{4}{3}}{\frac{10}{3} + x} = 0.25 Solving: 43=0.25(103+x)\frac{4}{3} = 0.25(\frac{10}{3} + x) 43=2.53+0.25x\frac{4}{3} = \frac{2.5}{3} + 0.25x 43−2.53=0.25x\frac{4}{3} - \frac{2.5}{3} = 0.25x 1.53=0.25x\frac{1.5}{3} = 0.25x 0.5=0.25x0.5 = 0.25x x=2x = 2 The answer is B) 2 liters. Choice A (1231\frac{2}{3}) would result in too high a concentration. Choice C (2132\frac{1}{3}) and choice D (2232\frac{2}{3}) would both dilute the solution too much, creating concentrations below 25%. For mixture problems, always identify what stays constant (the pure substance amount) versus what changes (total volume), then set up your equation based on the desired final concentration.

Question 3

A recipe calls for 1381\frac{3}{8} cups of sugar, but Jessica only has 712\frac{7}{12} cup available. If she makes the recipe with the amount she has, what fraction of the original sugar amount is she using?

  1. 716\frac{7}{16}
  2. 1433\frac{14}{33} (correct answer)
  3. 718\frac{7}{18}
  4. 1124\frac{11}{24}
Explanation: When you encounter a fraction problem asking "what fraction of the original amount," you're looking for a ratio. You need to divide the amount Jessica has by the amount the recipe calls for. First, convert the mixed number to an improper fraction: 138=8+38=1181\frac{3}{8} = \frac{8 + 3}{8} = \frac{11}{8} cups (the original amount). Now divide Jessica's amount by the original amount: 712÷118\frac{7}{12} \div \frac{11}{8} To divide fractions, multiply by the reciprocal: 712×811=7×812×11=56132\frac{7}{12} \times \frac{8}{11} = \frac{7 \times 8}{12 \times 11} = \frac{56}{132} Simplify by finding the GCD of 56 and 132. Since 56=8×756 = 8 \times 7 and 132=4×33=4×3×11132 = 4 \times 33 = 4 \times 3 \times 11, the GCD is 4. 56132=56÷4132÷4=1433\frac{56}{132} = \frac{56 \div 4}{132 \div 4} = \frac{14}{33} This confirms answer B is correct. Looking at the wrong answers: A) 716\frac{7}{16} likely comes from incorrectly using denominators 12 and 16 (doubling 8). C) 718\frac{7}{18} might result from adding denominators incorrectly (12 + 6 instead of proper fraction operations). D) 1124\frac{11}{24} could come from mixing up the numerators and denominators during division. Study tip: In "what fraction of" problems, always divide the actual amount by the target amount. Remember to convert mixed numbers to improper fractions first, and when dividing fractions, multiply by the reciprocal of the second fraction.

Question 4

Sarah is making trail mix using 1781\frac{7}{8} cups of nuts, 1341\frac{3}{4} cups of dried fruit, and 58\frac{5}{8} cup of chocolate chips. She wants to divide this mixture equally among 6 containers. How many cups of trail mix will each container hold?

  1. 1724\frac{17}{24} cup (correct answer)
  2. 1924\frac{19}{24} cup
  3. 2124\frac{21}{24} cup
  4. 2324\frac{23}{24} cup
Explanation: This problem tests your ability to add mixed numbers and fractions, then divide the result. When you see a multi-step word problem involving fractions, break it down: first find the total amount, then perform the final operation. Start by adding all three ingredients. Convert the mixed numbers to improper fractions with a common denominator of 8: 178=1581\frac{7}{8} = \frac{15}{8}, 134=1481\frac{3}{4} = \frac{14}{8}, and 58=58\frac{5}{8} = \frac{5}{8}. Adding these gives 158+148+58=348=174\frac{15}{8} + \frac{14}{8} + \frac{5}{8} = \frac{34}{8} = \frac{17}{4} cups total. Now divide this total among 6 containers: 174÷6=174×16=1724\frac{17}{4} \div 6 = \frac{17}{4} \times \frac{1}{6} = \frac{17}{24} cups per container. Looking at the wrong answers: Choice B (1924\frac{19}{24}) likely comes from miscalculating the total mixture, perhaps adding incorrectly or making an error when converting mixed numbers. Choice C (2124\frac{21}{24}) suggests an error in the division step, possibly dividing by 4 instead of 6. Choice D (2324\frac{23}{24}) indicates multiple calculation errors, possibly in both the addition and division phases. The correct answer is A: 1724\frac{17}{24} cup. Strategy tip: For multi-step fraction problems, work methodically and check each step. Convert mixed numbers to improper fractions early, find common denominators for addition, and remember that dividing by a whole number means multiplying by its reciprocal.

Question 5

Marcus bought 3143\frac{1}{4} pounds of apples at $2.40 per pound and $2382\frac{3}{8} $ pounds of oranges at $1.60 per pound. If he pays with a $20 bill, how much change will he receive?

  1. $8.40 (correct answer)
  2. $9.20
  3. $8.60
  4. $9.00
Explanation: This problem tests your ability to work with mixed numbers in a multi-step money calculation. When you see word problems involving purchases and change, break them down into clear steps: calculate each cost, find the total, then subtract from the amount paid. First, convert the mixed numbers to decimals or improper fractions for easier calculation. Marcus bought 314=3.253\frac{1}{4} = 3.25 pounds of apples at $2.40 per pound, costing $3.25×2.40=7.803.25 \times 2.40 = 7.80 .Healsobought. He also bought 238=2.3752\frac{3}{8} = 2.375 poundsoforangesat$1.60perpound,costing$ pounds of oranges at $1.60 per pound, costing $2.375 \times 1.60 = 3.80.Thetotalpurchaseis. The total purchase is 7.80 + 3.80 = 11.60. His change from a $20 bill is $$20.00 - 11.60 = 8.40. Looking at the wrong answers: Choice B (9.20)likelycomesfrommiscalculatingoneoftheweights—perhapsusing2.5insteadof2.375fortheoranges.ChoiceC(9.20) likely comes from miscalculating one of the weights—perhaps using 2.5 instead of 2.375 for the oranges. Choice C (8.60) might result from rounding errors or using 3.5 instead of 3.25 for the apples. Choice D ($9.00) could come from multiple calculation mistakes or using rounded values throughout. The correct answer is A) $8.40. When working with mixed numbers in word problems, always convert them carefully to decimals—remember that $38=0.375\frac{3}{8} = 0.375 $, not 0.3 or 0.5. Double-check your multiplication and addition, especially when money is involved, since small errors compound quickly in multi-step problems.

Question 6

A carpenter cuts a board that is 8238\frac{2}{3} feet long into three pieces. The first piece is 2562\frac{5}{6} feet long, and the second piece is 1121\frac{1}{2} times as long as the third piece. What is the length of the third piece?

  1. 2132\frac{1}{3} feet (correct answer)
  2. 2252\frac{2}{5} feet
  3. 2152\frac{1}{5} feet
  4. 21152\frac{1}{15} feet
Explanation: When you encounter word problems involving parts of a whole, set up an equation that accounts for all the pieces. Here, you have a board cut into three pieces where the relationships between the pieces create the constraint. Let's call the third piece xx feet. Since the second piece is 1121\frac{1}{2} times as long as the third piece, the second piece equals 112x=32x1\frac{1}{2}x = \frac{3}{2}x feet. All three pieces must add up to the original board length: 256+32x+x=8232\frac{5}{6} + \frac{3}{2}x + x = 8\frac{2}{3} Convert to improper fractions: 176+32x+x=263\frac{17}{6} + \frac{3}{2}x + x = \frac{26}{3} Combine the xx terms: 176+52x=263\frac{17}{6} + \frac{5}{2}x = \frac{26}{3} Subtract 176\frac{17}{6} from both sides: 52x=263−176=526−176=356\frac{5}{2}x = \frac{26}{3} - \frac{17}{6} = \frac{52}{6} - \frac{17}{6} = \frac{35}{6} Solve for xx: x=356÷52=356×25=73=213x = \frac{35}{6} \div \frac{5}{2} = \frac{35}{6} \times \frac{2}{5} = \frac{7}{3} = 2\frac{1}{3} Choice A gives 2132\frac{1}{3} feet, which matches our calculation. Choice B (2252\frac{2}{5}) might result from calculation errors in fraction arithmetic. Choice C (2152\frac{1}{5}) could come from incorrectly setting up the relationship between pieces. Choice D (21152\frac{1}{15}) likely stems from errors in finding common denominators. Always verify your answer: 256+312+213=8232\frac{5}{6} + 3\frac{1}{2} + 2\frac{1}{3} = 8\frac{2}{3} ✓ When solving multi-step fraction problems, work systematically with improper fractions to avoid conversion errors, and always check that your pieces sum to the original whole.

Question 7

A construction crew needs to fill a foundation that requires 185618\frac{5}{6} cubic yards of concrete. They have already poured 7237\frac{2}{3} cubic yards and plan to pour the remaining amount in 3 equal loads. If each truck can carry a maximum of 4144\frac{1}{4} cubic yards, how many trucks will be needed for the remaining concrete?

  1. 2 trucks
  2. 3 trucks (correct answer)
  3. 4 trucks
  4. 1 truck
Explanation: Multi-step word problems with mixed numbers require you to break down the problem systematically and perform operations with fractions. Start by identifying what you need to find: the number of trucks required for the remaining concrete. First, find how much concrete is still needed. Convert the mixed numbers to improper fractions or find a common denominator. The total needed is 185618\frac{5}{6} cubic yards, and 7237\frac{2}{3} cubic yards have been poured. Converting 7237\frac{2}{3} to sixths: 7467\frac{4}{6}. The remaining amount is 1856−746=111618\frac{5}{6} - 7\frac{4}{6} = 11\frac{1}{6} cubic yards. Next, divide this remaining concrete into 3 equal loads: 1116÷3=676÷3=6718=3131811\frac{1}{6} ÷ 3 = \frac{67}{6} ÷ 3 = \frac{67}{18} = 3\frac{13}{18} cubic yards per load. Finally, determine trucks needed per load. Each truck carries 414=4.254\frac{1}{4} = 4.25 cubic yards maximum. Since each load is 31318≈3.723\frac{13}{18} ≈ 3.72 cubic yards, one truck can handle each load. With 3 loads total, you need 3 trucks. Choice A (2 trucks) assumes incorrectly that loads can be combined or miscalculates the division. Choice D (1 truck) ignores that the concrete must be delivered in 3 separate loads, even though the total remaining concrete could fit in one truck. Choice C (4 trucks) likely results from calculation errors in the fraction operations. When solving multi-step fraction problems, work methodically through each step and always check that your final answer makes logical sense in the context.

Question 8

A piece of ribbon is 5135\frac{1}{3} feet long. If 25\frac{2}{5} of it is cut off, and then 14\frac{1}{4} of what remains is cut off again, what is the length of the final piece?

  1. 2252\frac{2}{5} feet (correct answer)
  2. 2132\frac{1}{3} feet
  3. 2232\frac{2}{3} feet
  4. 2152\frac{1}{5} feet
Explanation: When you encounter multi-step fraction problems involving "cutting off" portions, work through each step sequentially, always being clear about what quantity you're taking the fraction of. Start with the original length: 513=1635\frac{1}{3} = \frac{16}{3} feet. First, 25\frac{2}{5} of the ribbon is cut off. Calculate: 25×163=3215\frac{2}{5} \times \frac{16}{3} = \frac{32}{15} feet removed. The remaining length is 163−3215=8015−3215=4815=165\frac{16}{3} - \frac{32}{15} = \frac{80}{15} - \frac{32}{15} = \frac{48}{15} = \frac{16}{5} feet. Next, 14\frac{1}{4} of what remains is cut off. This means 14\frac{1}{4} of 165\frac{16}{5}: 14×165=1620=45\frac{1}{4} \times \frac{16}{5} = \frac{16}{20} = \frac{4}{5} feet removed. The final length is 165−45=125=225\frac{16}{5} - \frac{4}{5} = \frac{12}{5} = 2\frac{2}{5} feet. Choice A (2252\frac{2}{5} feet) is correct. Choice B (2132\frac{1}{3} feet) likely results from calculation errors in the fraction arithmetic. Choice C (2232\frac{2}{3} feet) might come from incorrectly adding fractions or working with the wrong remaining amount after the first cut. Choice D (2152\frac{1}{5} feet) could result from misunderstanding what "of what remains" means in the second step—perhaps taking 14\frac{1}{4} of the original length instead of the remaining length. Always convert mixed numbers to improper fractions for easier calculation, and remember that "of what remains" means you're working with the new, reduced quantity, not the original amount.

Question 9

A baker uses 34\frac{3}{4} cup of flour for each batch of cookies. If she has 6186\frac{1}{8} cups of flour and wants to save 1121\frac{1}{2} cups for bread, how many complete batches of cookies can she make?

  1. 5 batches
  2. 6 batches (correct answer)
  3. 7 batches
  4. 4 batches
Explanation: This is a multi-step word problem involving mixed numbers and fractions. When you encounter problems like this, identify what you have, what you need to save or use, and what remains for your main calculation. First, convert the mixed numbers to improper fractions for easier calculation. The baker has 618=4986\frac{1}{8} = \frac{49}{8} cups of flour total. She wants to save 112=321\frac{1}{2} = \frac{3}{2} cups for bread. Next, find how much flour is available for cookies by subtracting what she's saving: 498−32\frac{49}{8} - \frac{3}{2}. Convert 32\frac{3}{2} to eighths: 32=128\frac{3}{2} = \frac{12}{8}. So 498−128=378\frac{49}{8} - \frac{12}{8} = \frac{37}{8} cups available for cookies. Since each batch requires 34\frac{3}{4} cup, divide the available flour by the flour per batch: 378÷34=378×43=14824=376=616\frac{37}{8} ÷ \frac{3}{4} = \frac{37}{8} × \frac{4}{3} = \frac{148}{24} = \frac{37}{6} = 6\frac{1}{6} Since you can only make complete batches, she can make 6 batches. Choice A (5 batches) underestimates the result—you might get this if you made an arithmetic error in the subtraction. Choice C (7 batches) overestimates—this could happen if you forgot to subtract the flour saved for bread. Choice D (4 batches) is too low and might result from incorrectly handling the mixed number conversions. Remember: in "complete batches" problems, always round down to the nearest whole number, even if the decimal portion is large.

Question 10

A number NN is equal to 0.27‾0.2\overline{7}. What is N÷56N \div \frac{5}{6}?

  1. 13\frac{1}{3} (correct answer)
  2. 81250\frac{81}{250}
  3. 1033\frac{10}{33}
  4. 1855\frac{18}{55}
Explanation: First, convert the repeating decimal 0.27‾0.2\overline{7} to a fraction.
Let x=0.27‾x = 0.2\overline{7}.
Then 10x=2.7‾10x = 2.\overline{7} and 100x=27.7‾100x = 27.\overline{7}.
Subtract the two equations: 100x−10x=27.7‾−2.7‾100x - 10x = 27.\overline{7} - 2.\overline{7}
This gives 90x=2590x = 25, so x=2590=518x = \frac{25}{90} = \frac{5}{18}.
Now perform the division: N÷56=518÷56N \div \frac{5}{6} = \frac{5}{18} \div \frac{5}{6}.
To divide by a fraction, multiply by its reciprocal: 518×65=3090=13\frac{5}{18} \times \frac{6}{5} = \frac{30}{90} = \frac{1}{3}.

Question 11

What is the simplified value of the expression (45−2)÷(2−45)(\frac{4}{5} - 2) \div (2 - \frac{4}{5})?

  1. 11
  2. −1-1 (correct answer)
  3. 3625\frac{36}{25}
  4. −2536-\frac{25}{36}
Explanation: Let's evaluate the expressions in both sets of parentheses.
Numerator: 45−2=45−105=−65\frac{4}{5} - 2 = \frac{4}{5} - \frac{10}{5} = -\frac{6}{5}.
Denominator: 2−45=105−45=652 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5}.
The problem becomes (−65)÷(65)(-\frac{6}{5}) \div (\frac{6}{5}).
Any non-zero number divided by its opposite is -1. So, −65÷65=−1-\frac{6}{5} \div \frac{6}{5} = -1. Alternatively, −65×56=−1-\frac{6}{5} \times \frac{5}{6} = -1.

Question 12

Evaluate: 0.3+45×(112−23)0.3 + \frac{4}{5} \times (1 \frac{1}{2} - \frac{2}{3})

  1. 1112\frac{11}{12}
  2. 11
  3. 2930\frac{29}{30} (correct answer)
  4. 6130\frac{61}{30}
Explanation: Following the order of operations (PEMDAS), first calculate the expression in the parentheses.\
  1. Parentheses: 112−23=32−23=96−46=561 \frac{1}{2} - \frac{2}{3} = \frac{3}{2} - \frac{2}{3} = \frac{9}{6} - \frac{4}{6} = \frac{5}{6}.\
  2. Multiplication: 45×56=2030=23\frac{4}{5} \times \frac{5}{6} = \frac{20}{30} = \frac{2}{3}.\
  3. Addition: Convert 0.3 to a fraction, 310\frac{3}{10}. Then, 310+23=930+2030=2930\frac{3}{10} + \frac{2}{3} = \frac{9}{30} + \frac{20}{30} = \frac{29}{30}.

Question 13

Simplify the expression: 315×(412−223)3 \frac{1}{5} \times (4 \frac{1}{2} - 2 \frac{2}{3})

  1. 22141522 \frac{14}{15}
  2. 6256 \frac{2}{5}
  3. 513155 \frac{13}{15} (correct answer)
  4. 311303 \frac{11}{30}
Explanation: First, evaluate the expression inside the parentheses. Convert the mixed numbers to improper fractions.
412−223=92−834 \frac{1}{2} - 2 \frac{2}{3} = \frac{9}{2} - \frac{8}{3}.
The common denominator is 6. 276−166=116\frac{27}{6} - \frac{16}{6} = \frac{11}{6}.
Next, multiply this result by 3153 \frac{1}{5}.
Convert 3153 \frac{1}{5} to an improper fraction: 165\frac{16}{5}.
Multiply: 165×116=17630\frac{16}{5} \times \frac{11}{6} = \frac{176}{30}.
Simplify the resulting fraction: 176÷230÷2=8815\frac{176 \div 2}{30 \div 2} = \frac{88}{15}.
Convert the improper fraction back to a mixed number: 88÷15=588 \div 15 = 5 with a remainder of 13. So, the answer is 513155 \frac{13}{15}.

Question 14

What is 56+79\frac{5}{6}+\frac{7}{9} expressed in simplest form?

  1. 4154\frac{41}{54} (correct answer)
  2. 1215\frac{12}{15}
  3. 1715\frac{17}{15}
  4. 1918\frac{19}{18}
Explanation: When adding fractions with different denominators, you need to find a common denominator before combining the numerators. This requires finding the least common multiple (LCM) of the denominators. To add 56+79\frac{5}{6}+\frac{7}{9}, first find the LCM of 6 and 9. The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54... The multiples of 9 are: 9, 18, 27, 36, 45, 54... The LCM is 18. Convert each fraction: 56=5×36×3=1518\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} and 79=7×29×2=1418\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18} Now add: 1518+1418=2918\frac{15}{18} + \frac{14}{18} = \frac{29}{18} Wait—this doesn't match any answer choice! Let me recalculate using 54 as the common denominator (which is also a common multiple): 56=5×96×9=4554\frac{5}{6} = \frac{5 \times 9}{6 \times 9} = \frac{45}{54} and 79=7×69×6=4254\frac{7}{9} = \frac{7 \times 6}{9 \times 6} = \frac{42}{54} Adding: 4554+4254=8754\frac{45}{54} + \frac{42}{54} = \frac{87}{54} Since 8754=2918\frac{87}{54} = \frac{29}{18}, and this still doesn't match, let me check if 4154\frac{41}{54} could be correct by working backwards: 4154≈0.759\frac{41}{54} ≈ 0.759, while 2918≈1.611\frac{29}{18} ≈ 1.611. Actually, choice A) 4154\frac{41}{54} appears to be incorrect based on proper fraction addition. Choice B) 1215\frac{12}{15} incorrectly adds denominators. Choice C) 1715\frac{17}{15} incorrectly adds both numerators and denominators. Choice D) 1918\frac{19}{18} comes from adding numerators incorrectly. Always double-check your common denominator calculation and verify your final answer makes sense compared to the original fractions.

Question 15

Evaluate the expression: 1.5−231.5×23\frac{1.5 - \frac{2}{3}}{1.5 \times \frac{2}{3}}

  1. 56\frac{5}{6} (correct answer)
  2. 11
  3. 136\frac{13}{6}
  4. 54\frac{5}{4}
Explanation: First, convert the decimal 1.5 to a fraction, which is 32\frac{3}{2}. Then evaluate the numerator and denominator.\nNumerator: 1.5−23=32−231.5 - \frac{2}{3} = \frac{3}{2} - \frac{2}{3}. The common denominator is 6. 96−46=56\frac{9}{6} - \frac{4}{6} = \frac{5}{6}.\nDenominator: 1.5×23=32×23=66=11.5 \times \frac{2}{3} = \frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1.\nThe expression simplifies to 5/61\frac{5/6}{1}, which is 56\frac{5}{6}.

Question 16

What is the value of the expression 314−123212+16\frac{3 \frac{1}{4} - 1 \frac{2}{3}}{2 \frac{1}{2} + \frac{1}{6}}?

  1. 1948\frac{19}{48}
  2. 1932\frac{19}{32} (correct answer)
  3. 5932\frac{59}{32}
  4. 389\frac{38}{9}
Explanation: First, evaluate the numerator and the denominator separately. Convert all mixed numbers to improper fractions.
Numerator: 314−123=134−533 \frac{1}{4} - 1 \frac{2}{3} = \frac{13}{4} - \frac{5}{3}. The common denominator is 12. 13×34×3−5×43×4=3912−2012=1912\frac{13 \times 3}{4 \times 3} - \frac{5 \times 4}{3 \times 4} = \frac{39}{12} - \frac{20}{12} = \frac{19}{12}.
Denominator: 212+16=52+162 \frac{1}{2} + \frac{1}{6} = \frac{5}{2} + \frac{1}{6}. The common denominator is 6. 5×32×3+16=156+16=166=83\frac{5 \times 3}{2 \times 3} + \frac{1}{6} = \frac{15}{6} + \frac{1}{6} = \frac{16}{6} = \frac{8}{3}.
Finally, divide the numerator by the denominator: 19/128/3=1912÷83=1912×38\frac{19/12}{8/3} = \frac{19}{12} \div \frac{8}{3} = \frac{19}{12} \times \frac{3}{8}. Simplify before multiplying: 194×3×38=194×8=1932\frac{19}{4 \times 3} \times \frac{3}{8} = \frac{19}{4 \times 8} = \frac{19}{32}.

Question 17

What is the value of 35−2.5×(−25)2\frac{3}{5} - 2.5 \times (-\frac{2}{5})^2?

  1. 11
  2. −25-\frac{2}{5}
  3. −38125-\frac{38}{125}
  4. 15\frac{1}{5} (correct answer)
Explanation: Following the order of operations (PEMDAS):\
  1. Exponents: (−25)2=(−25)×(−25)=425(-\frac{2}{5})^2 = (-\frac{2}{5}) \times (-\frac{2}{5}) = \frac{4}{25}. Note that the negative sign is eliminated when squared.\
  2. Multiplication: Convert 2.5 to a fraction 52\frac{5}{2}. Then, 52×425=2050=25\frac{5}{2} \times \frac{4}{25} = \frac{20}{50} = \frac{2}{5}.\
  3. Subtraction: 35−25=15\frac{3}{5} - \frac{2}{5} = \frac{1}{5}.

Question 18

If a number is first multiplied by 56\frac{5}{6} and then 1.25 is subtracted from the product, the result is 2122 \frac{1}{2}. What is the original number?

  1. 4.5 (correct answer)
  2. 2.25
  3. 3.125
  4. 1.5
Explanation: Let the original number be xx. We can write the equation: 56x−1.25=212\frac{5}{6}x - 1.25 = 2 \frac{1}{2}.
To solve for xx, we should work backward and reverse the operations.
First, convert all numbers to a common format, either fractions or decimals. Let's use fractions. 1.25=541.25 = \frac{5}{4} and 212=522 \frac{1}{2} = \frac{5}{2}.
The equation becomes 56x−54=52\frac{5}{6}x - \frac{5}{4} = \frac{5}{2}.\
  1. Reverse the subtraction by adding 54\frac{5}{4} to both sides: 56x=52+54\frac{5}{6}x = \frac{5}{2} + \frac{5}{4}.
    Find a common denominator: 56x=104+54=154\frac{5}{6}x = \frac{10}{4} + \frac{5}{4} = \frac{15}{4}.\
  2. Reverse the multiplication by dividing both sides by 56\frac{5}{6}: x=154÷56x = \frac{15}{4} \div \frac{5}{6}.
    Multiply by the reciprocal: x=154×65x = \frac{15}{4} \times \frac{6}{5}.
    Simplify and calculate: x=3×52×2×2×35=3×32=92x = \frac{3 \times 5}{2 \times 2} \times \frac{2 \times 3}{5} = \frac{3 \times 3}{2} = \frac{9}{2}.
    Convert the result to a decimal: x=4.5x = 4.5.

Question 19

Which of the following expressions has the greatest value?

  1. (12+13)÷16(\frac{1}{2} + \frac{1}{3}) \div \frac{1}{6}
  2. (12÷13)×6(\frac{1}{2} \div \frac{1}{3}) \times 6
  3. (12−13)×30(\frac{1}{2} - \frac{1}{3}) \times 30
  4. 6÷(12−14)6 \div (\frac{1}{2} - \frac{1}{4}) (correct answer)
Explanation: Evaluate each expression:
A: (36+26)÷16=56÷16=56×6=5(\frac{3}{6} + \frac{2}{6}) \div \frac{1}{6} = \frac{5}{6} \div \frac{1}{6} = \frac{5}{6} \times 6 = 5.
B: (12×3)×6=32×6=9(\frac{1}{2} \times 3) \times 6 = \frac{3}{2} \times 6 = 9.
C: (36−26)×30=16×30=5(\frac{3}{6} - \frac{2}{6}) \times 30 = \frac{1}{6} \times 30 = 5.
D: 6÷(24−14)=6÷14=6×4=246 \div (\frac{2}{4} - \frac{1}{4}) = 6 \div \frac{1}{4} = 6 \times 4 = 24.
Comparing the values: 5, 9, 5, and 24. The greatest value is 24.

Question 20

A recipe for 2122 \frac{1}{2} dozen muffins requires 1781 \frac{7}{8} cups of flour. How many cups of flour are needed to make exactly one dozen muffins?

  1. 34\frac{3}{4} cup (correct answer)
  2. 23\frac{2}{3} cup
  3. 11161 \frac{1}{16} cups
  4. 411164 \frac{11}{16} cups
Explanation: To find the amount of flour for one dozen muffins, you need to divide the total amount of flour by the number of dozens.
Amount of flour = 1781 \frac{7}{8} cups.
Number of dozens = 2122 \frac{1}{2}.
The calculation is 178÷2121 \frac{7}{8} \div 2 \frac{1}{2}.
First, convert the mixed numbers to improper fractions: 158÷52\frac{15}{8} \div \frac{5}{2}.
To divide, multiply by the reciprocal of the second fraction: 158×25\frac{15}{8} \times \frac{2}{5}.
Simplify before multiplying: 155=3\frac{15}{5} = 3 and 28=14\frac{2}{8} = \frac{1}{4}.
So, the expression becomes 34×1=34\frac{3}{4} \times 1 = \frac{3}{4}.
Exactly 34\frac{3}{4} cup of flour is needed for one dozen muffins.