All questions
Question 1
A library has 1,260 stickers to place equally on 35 reading charts. What is the quotient of 1,260÷35?
- 9
- 36 (correct answer)
- 45
- 360
Explanation: Division with two-digit divisors relies on place value to split the dividend, for instance, viewing 1,260 as 1,050 + 210 for dividing by 35. Estimating the quotient involves approximating, like 1,260 close to 1,225, where 1,225 ÷ 35 = 35, but actual calculation yields 36. Checking with multiplication, 35 × 36 = 1,260, confirms the quotient's accuracy. This ties into the area model of division, visualizing the breakdown into rectangles based on place value. One misconception is confusing partial quotients with the final answer, but adding them properly resolves it. Such reasoning ensures comprehensive handling of the dividend, fostering precise division. It also enhances problem-solving skills across mathematical contexts.
Question 2
A library has 1,176 books to place equally on 28 shelves. A student says, "I can check my division by multiplying the quotient by 28 to get 1,176." What is the quotient of 1,176÷28?
- 14
- 42 (correct answer)
- 84
- 420
Explanation: Division with two-digit divisors uses place value by decomposing the dividend, such as viewing 1,176 as 1,100 + 76, to facilitate easier division. For estimating 1,176 ÷ 28, calculate 28 × 40 = 1,120, which is close to 1,176, suggesting the quotient is around 40 and needs slight adjustment upward. Multiplication serves to check by multiplying the quotient back by 28 to confirm it equals 1,176, like 42 × 28 = 1,176. This connects to the partial products strategy, adding multiples such as 40 × 28 and 2 × 28 to reach the total. A misconception is mistaking the divisor for a single digit and ignoring the tens place, which can double the error in the quotient. Using place value reasoning promotes step-by-step accuracy in division tasks. It generalizes by building confidence in verifying answers through the multiplication-division connection.
Question 3
A coach has 864 water bottles to pack equally into boxes that each hold 24 bottles. Which statement gives the correct quotient and shows how multiplication can check the division 864÷24?
- The quotient is 360 because 24 goes into 864 about 36 times and then you add a zero.
- The quotient is 36 because 24×36=864, so multiplication checks the division. (correct answer)
- The quotient is 34 because 24×34=816 and that is close enough to 864.
- The quotient is 3,456 because 864×24=3,456.
Explanation: Division with two-digit divisors uses place value to break down the dividend into manageable parts, like hundreds and tens. Estimating the quotient involves finding how many times the divisor fits into larger place values, such as seeing that 24 goes into 720 thirty times since 24×30=720. Using multiplication to check means multiplying the estimated quotient by the divisor to verify if it equals the dividend, confirming 24×36=864. This connects to the partial quotients strategy, where you add quotients from each place value, like 30+6=36. A common misconception is adding an extra zero to the quotient, but that ignores the actual place value relationships. Reasoning with place value helps ensure the quotient is accurate by aligning the multiplication back to the original dividend. Overall, this approach builds confidence in division by linking it to familiar multiplication facts.
Question 4
A teacher has 864 pencils to pack equally into boxes that hold 24 pencils each. What is the quotient of 864÷24?
- 18
- 36 (correct answer)
- 72
- 360
Explanation: Division with two-digit divisors uses place value by breaking down the dividend into hundreds, tens, and ones to simplify the process. To estimate the quotient for 864 ÷ 24, note that 24 × 30 = 720, which is less than 864, and 24 × 40 = 960, which is too much, so the quotient is between 30 and 40. Using multiplication to check, multiply the proposed quotient by 24 and see if it equals 864, such as 36 × 24 = 864. This connects to the strategy of partial quotients, where you add groups like 30 groups of 24 (720) and then 6 more (144) to reach 864. A common misconception is forgetting to account for the remainder after the first partial quotient, leading to an underestimate. By applying place value reasoning, you ensure each part of the dividend is divided accurately, supporting precise calculations. This method generalizes to any division problem by reinforcing the inverse relationship between multiplication and division for verification.
Question 5
A science club has 672 beads to make bracelets. Each bracelet uses 21 beads. What is the quotient for 672÷21, and how does multiplication help check it?
- The quotient is 32 because 21×32=672, so multiplication checks the division. (correct answer)
- The quotient is 320 because 21 is two digits, so the quotient must have a zero.
- The quotient is 21 because 672÷21 means 21÷672.
- The quotient is 40 because 672÷20≈33.6, so rounding up gives 40.
Explanation: Division with two-digit divisors uses place value to understand how the divisor interacts with each digit group in the dividend, such as tens and ones. Estimating the quotient involves testing multiples close to the dividend, like 21×30=630 for 672. Using multiplication to check means computing quotient times divisor to match the dividend, confirming 21×32=672. This connects to breaking the quotient into parts, like 30+2, and adding the products. A misconception is assuming the quotient must include a zero just because the divisor has two digits, ignoring actual calculation. Place value reasoning helps by aligning the multiplication to the dividend's structure for accuracy. In general, this supports precise division by emphasizing verification through inverse operations.
Question 6
A coach has 1,152 water bottles to share equally among 32 players. A student estimates 1,152≈1,280, and since 32×40=1,280, the quotient should be close to 40. What is the quotient of 1,152÷32?
- 36 (correct answer)
- 360
- 48
- 32
Explanation: Division with two-digit divisors uses place value to divide larger numbers by considering their expanded form, like treating 1,152 as 1,000 + 152. Estimating the quotient for 1,152 ÷ 32 involves approximating 1,152 as 1,280, and since 32 × 40 = 1,280, the exact quotient should be near 40 but adjusted lower. Multiplication checks the result by verifying if the quotient times 32 equals 1,152, for example, 36 × 32 = 1,152. This relates to the long division algorithm, where you divide step-by-step into the hundreds and then the remaining tens and ones. One misconception is over-relying on estimates without exact calculation, which might lead to accepting 40 instead of adjusting to 36. Place value reasoning helps break down complex divisions into manageable parts, ensuring accuracy. Overall, this approach supports reliable division by linking estimation, calculation, and verification through multiplication.
Question 7
A music teacher has 1,680 rhythm cards and puts them into equal stacks of 42 cards. What is the quotient of 1,680÷42?
- 4
- 40 (correct answer)
- 84
- 28
Explanation: Division with two-digit divisors incorporates place value to decompose, like 1,680 as 1,260 + 420 for dividing by 42. Estimating, 1,680 ÷ 42 is around 40, as 42 × 40 = 1,680 exactly. Check with multiplication: 42 × 40 = 1,680, confirming it. This connects to mental math strategies using place value multiples. A misconception is dividing only leading digits, but full breakdown avoids it. Reasoning via place value leads to precise quotients. It builds foundational skills for advanced division.
Question 8
A science club has 1,596 beads to make bracelets. Each bracelet uses 28 beads. What is the quotient of 1,596÷28?
- 57 (correct answer)
- 28
- 75
- 19
Explanation: Division with two-digit divisors utilizes place value to partition the dividend, like 1,596 as 1,400 + 196 for division by 28. Estimating involves noting 1,596 close to 1,400, where 1,400 ÷ 28 = 50, with further calculation giving 57. You check via multiplication: 28 × 57 = 1,596, confirming correctness. This links to the box method, dividing based on place value sections. Misconception often arises from mishandling remainders in place value breaks, but verification fixes it. Reasoning this way ensures thorough accuracy in division tasks. It generalizes to efficient problem-solving in math.
Question 9
A student says, "1,248÷26=48 because 26×48=1,248." Another student says, "No, the quotient is 480 because 26 goes into 124 about 4 times, so add a zero." Which claim about 1,248÷26 is incorrect?
- The claim that the quotient is 48 is incorrect.
- The claim that the quotient is 480 is incorrect. (correct answer)
- Both claims are correct because division and multiplication are related.
- Neither claim can be checked with multiplication.
Explanation: Division with two-digit divisors uses place value to analyze claims, like evaluating 1,248 ÷ 26 by breaking down the numbers. Estimating the quotient, 1,248 is close to 1,300, and 1,300 ÷ 26 ≈ 50, pointing away from 480 toward 48. Checking with multiplication, 26 × 48 = 1,248, while 26 × 480 = 12,480, shows 480 is incorrect. This connects to error analysis strategies, using place value to identify mistakes in reasoning. A common misconception is appending zeros without considering place value, leading to inflated quotients like 480. Such reasoning helps distinguish correct from incorrect claims, ensuring accurate division. It generalizes to critical thinking in mathematical verification.
Question 10
A school bought 1,152 pencils and packed them equally into boxes that hold 24 pencils each. What is the quotient of 1,152÷24?
- 96
- 48 (correct answer)
- 72
- 24
Explanation: Division with two-digit divisors uses place value to break down the dividend, such as thinking of 1,152 as 960 + 192, making it easier to divide by 24. To estimate the quotient, note that 1,152 is close to 1,200, and 1,200 ÷ 24 = 50, so the answer should be near 50, but adjusting for the actual numbers points to 48. You can check the division by multiplying the quotient by the divisor, like 24 × 48 = 1,152, confirming it's exact. This connects to the partial quotients strategy, where you divide parts like 960 ÷ 24 = 40 and 192 ÷ 24 = 8, adding to 48. A common misconception is ignoring remainders when breaking down place values, which can lead to incomplete quotients. Using place value reasoning ensures all parts of the number are accounted for, promoting accuracy in division. This method strengthens overall number sense, helping with more complex problems.
Question 11
A band has 1,540 flyers to hand out equally to 28 students. Use estimation to think about the size of the quotient and then use multiplication to check the exact answer. What is the quotient of 1,540÷28?
- 55 (correct answer)
- 45
- 550
- 28
Explanation: Division with two-digit divisors uses place value by considering the positions in numbers like 1,540 to divide efficiently. For estimation in 1,540 ÷ 28, 28 × 50 = 1,400 is less, and adjusting to 28 × 55 = 1,540 fits exactly. Multiplication confirms the quotient by checking if it times 28 equals 1,540. This connects to the scaffold method, building the quotient incrementally. A misconception is scaling up the quotient unnecessarily, leading to answers like 550. Place value reasoning ensures accurate scaling and adjustment in division. It generalizes by linking estimation to multiplication for reliable results in various contexts.
Question 12
A school orders 3,276 pencils for the year. The pencils come in packages of 84 pencils each. The school wants to distribute all complete packages equally among 13 classrooms. How many packages will each classroom receive?
- 3 packages (correct answer)
- 4 packages
- 2 packages
- 5 packages
Explanation: First, find how many complete packages there are: 3,276 ÷ 84 = 39 packages. Then divide the packages among classrooms: 39 ÷ 13 = 3 packages per classroom. Choice B assumes incorrect calculation of total packages. Choice C results from errors in the two-step division process. Choice D represents miscalculation in either the first or second division step.
Question 13
A librarian has 1,935 books to arrange on shelves. Each shelf holds exactly 43 books. After filling complete shelves, she finds there are some books left over that don't fill a complete shelf. How many complete shelves can she fill?
- 44 shelves
- 45 shelves (correct answer)
- 46 shelves
- 43 shelves
Explanation: Divide 1,935 by 43: 1,935 ÷ 43 = 45 remainder 0. This means exactly 45 complete shelves can be filled with no books left over. Choice A results from calculation errors in long division. Choice C represents mistakes in the division algorithm. Choice D occurs from misunderstanding the quotient in the division process.
Question 14
A cafeteria has 1,008 apples to place equally into crates. Each crate holds 28 apples. A student checks with multiplication: 28×30=840 and 28×6=168, and 840+168=1,008. How does multiplication help check the result of 1,008÷28?
- It helps because you can multiply 28 by 36 to get 1,008, showing the quotient is 36. (correct answer)
- It helps because you can multiply 1,008 by 28 to get the quotient.
- It helps because you can multiply 28 by 3.6 to get 100.8, so the quotient is 3.6.
- It helps because you can multiply 28 by 360 to get 1,008.
Explanation: Division with two-digit divisors uses place value to handle the dividend in chunks, aligning with the divisor's size. Estimating the quotient starts with base multiples, like 28×30=840 for 1,008. Using multiplication to check means multiplying quotient by divisor to recover the dividend, showing 28×36=1,008. This connects to breaking down the problem into verifiable steps, similar to an area model. A misconception is reversing the multiplication to find the quotient incorrectly, like multiplying dividend by divisor. Place value reasoning aids accuracy by ensuring parts fit the whole. This general approach builds precise division skills through inverse checking.
Question 15
A class has 1,512 crayons to share equally among 36 students. Use estimation: 1,512 is close to 1,440, and 1,440÷36=40, so the quotient should be a little more than 40. Division can be checked with multiplication: 36×quotient=1,512. What is the quotient of 1,512÷36?
- 54
- 14
- 42 (correct answer)
- 420
Explanation: Division with two-digit divisors leverages place value for equitable sharing, as in 1,512 ÷ 36. To estimate, 1,512 is near 1,440, with 1,440 ÷ 36 = 40, refining to 42. Multiplication checks: 36 × 42 = 1,512, affirming the quotient. This ties to the ratio table model, scaling with place value. Misconception includes ignoring adjustments post-estimation, but verification prevents errors. Place value reasoning supports exact division outcomes. It enhances overall accuracy in computational tasks.
Question 16
A class has 1,296 centimeter cubes to build equal towers. Each tower uses 27 cubes. What is the quotient of 1,296÷27?
- 36
- 48 (correct answer)
- 480
- 27
Explanation: Division with two-digit divisors uses place value to decompose dividends like 1,296 into easier parts for division. Estimating 1,296 ÷ 27, 27 × 40 = 1,080, then adding 27 × 8 = 216 reaches 1,296, suggesting 48. Multiplication checks by verifying 48 × 27 = 1,296. This ties to the partial quotients strategy, adding groups step-by-step. One misconception is inverting the numbers, like dividing 27 by 1,296. Reasoning with place value supports precise group counting in division. It generalizes by using multiplication to validate and refine estimates effectively.
Question 17
A coach has 1,344 water bottles to share equally among 32 players. Estimation can help: 1,344 is close to 1,280, and 1,280÷32=40. Division can also be explained using multiplication (the quotient should make 32×quotient=1,344). What is the quotient of 1,344÷32?
- 42 (correct answer)
- 84
- 24
- 64
Explanation: Division with two-digit divisors uses place value to decompose the dividend, aiding in calculations like 1,344 ÷ 32. To estimate the quotient, recognize that 1,344 is close to 1,280, and 1,280 ÷ 32 = 40, with adjustments leading to 42 as the precise value. You can verify by multiplying back, such as 32 × 42 = 1,344, ensuring the result matches the dividend. This relates to the long division algorithm, where place value guides each step of estimating and subtracting. A frequent misconception is underestimating due to rounding errors, but refining the estimate corrects this. Place value reasoning supports breaking problems into simpler parts, leading to reliable quotients. Ultimately, this builds confidence in division through logical verification.
Question 18
A garden club has 1,584 seeds and wants to plant them in equal rows with 33 seeds in each row. Estimation: 1,584 is close to 1,650, and 1,650÷33=50, so the quotient should be a little less than 50. Use multiplication to check: 33×quotient=1,584. What is the quotient of 1,584÷33?
- 48 (correct answer)
- 16
- 480
- 33
Explanation: Division with two-digit divisors applies place value for organization, such as in 1,584 ÷ 33. Estimation: 1,584 near 1,650, 1,650 ÷ 33 = 50, adjusting to 48. Verify by multiplying: 33 × 48 = 1,584, ensuring accuracy. This relates to the long division process, emphasizing place value at each digit. Common misconception is over-rounding without correction, but multiplication checks it. Place value reasoning guarantees complete and correct division. It promotes effective strategies for real-world problems.
Question 19
A school orders 1,344 stickers and packs them equally into bags of 21 stickers each. Use an estimate to think about the size of the quotient and then use multiplication to check your exact answer. What is the quotient of 1,344÷21?
- 64 (correct answer)
- 84
- 640
- 168
Explanation: Division with two-digit divisors uses place value to handle the dividend by its digit positions, making large numbers more approachable. Estimating for 1,344 ÷ 21, 21 × 60 = 1,260 is less than 1,344, and 21 × 70 = 1,470 is too high, so the quotient is in the 60s. Multiplication checks the exact quotient by ensuring it times 21 equals 1,344, as in 64 × 21 = 1,344. This ties into the area model, where you visualize rectangles of 21 units wide to fit into 1,344. A common misconception is adding extra zeros to the quotient when dealing with even divisors, leading to answers like 640. Place value reasoning ensures each step accounts for the correct magnitude, aiding precision. This strategy generalizes to support accurate division by integrating estimation with multiplicative verification.
Question 20
A bakery has 1,248 cookies to pack equally into trays. Each tray holds 26 cookies.
A student breaks 1,248 into 1,040+208 and says, "Since 26×40=1,040 and 26×8=208, the quotient is 40+8." This uses place value and the idea that division can be explained using multiplication relationships.
Which strategy correctly divides the numbers?
- Divide 1,248 by 2 and then divide by 6 because 26=2+6.
- Use 26×40 and 26×8 to make 1,248, so the quotient is 48. (correct answer)
- Treat 26 like 2 and compute 1,248÷2=624, so the quotient is 624.
- Because 1,248 is close to 1,300, choose 50 without checking by multiplication.
Explanation: Division with two-digit divisors uses place value to break down numbers like 1,248 into components for accurate strategies. Estimating 1,248 ÷ 26, 26 × 40 = 1,040 and 26 × 8 = 208 sum to 1,248, identifying the correct approach as adding partial quotients. Multiplication verifies strategies by ensuring the product matches the dividend. This connects to the distributive property in division. A misconception is adding divisor parts instead of multiplying factors. Place value reasoning helps select valid methods for division. It generalizes by fostering strategy evaluation through estimation and checking.