Divide Multi-Digit Numbers - 6th Grade Math

A store has $2{,}019$ stickers and packs them into $16$ identical packs. Divide $2{,}019\div 16$ using the standard long division algorithm. What is the quotient with remainder?
Answer: $126\ \text{R}3$ This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 2,019 ÷ 16, divide 16 into 20, which goes 1 time (16×1=16), subtract 20-16=4, bring down 1 to make 41; 16 into 41 goes 2 times (16×2=32), subtract 41-32=9, bring down 9 to make 99; 16 into 99 goes 6 times (16×6=96), subtract 99-96=3, quotient 126 R3. Verify: 126×16 +3=2,016+3=2,019, matches. The correct answer is 126 R3, with 3 stickers left over. Mistakes often stem from wrong trial multiplication or forgetting to bring down. Ensure remainder < divisor. This represents packing items equally with possible leftovers.
A club collected $957 to share equally among $12$ students. Using long division, how much money does each student get, and what is the remainder?
Answer: $79\text{ R }9$ This question tests dividing multi-digit numbers using the standard long division algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 957 ÷ 12, divide 12 into 95 (7 times since 12×7=84 ≤95 <12×8=96), write 7, multiply 84, subtract 95-84=11, bring down 7 to make 117; 12 into 117 goes 9 times (12×9=108), subtract 117-108=9, so quotient 79 with remainder 9. Verify: 79×12 +9=948+9=957, which matches. Each student gets $79 with $9 left over, as in option A. Common mistakes include choosing a remainder ≥ divisor, like R12 or R21, which indicates an error in selecting the quotient digit. At each step, trial multiplication helps: check if divisor times digit fits without exceeding. This sharing money context shows practical use, and verification confirms the work.
A science club has $3{,}275$ beads to make bracelets. They split them equally among $25$ students. Divide $3{,}275\div 25$ using the standard long division algorithm. How many beads does each student get?
Answer: 131 This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 3,275 ÷ 25, divide 25 into 32, which goes 1 time (25×1=25), subtract 32-25=7, bring down 7 to make 77; 25 into 77 goes 3 times (25×3=75), subtract 77-75=2, bring down 5 to make 25; 25 into 25 goes 1 time (25×1=25), subtract 0, so quotient is 131. Verify: 131×25=3,275, exact match. Each student gets 131 beads. Common errors include forgetting to bring down digits or wrong subtraction, like 77-75=3. Ensure each step's remainder is less than the divisor before proceeding. This division applies to fair distribution, such as sharing beads among students.
A school store has 457 pencils to put into packs of 12. Using the standard long division algorithm, what is $457\div 12$ expressed as a quotient with a remainder?
Answer: $38\text{ R }1$ This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, repeat until complete, with quotient and remainder. For 457 ÷ 12, divide 12 into 45 (goes 3 times, 12×3=36≤45<12×4=48), write 3, multiply 36, subtract 45-36=9, bring down 7 to make 97; 12 into 97 goes 8 times (12×8=96≤97<12×9=108), multiply 96, subtract 97-96=1, so quotient 38 with remainder 1. Verify: 38×12 +1 =456+1=457, matching the dividend. The correct answer is 38 R1, as in choice A. Errors might include wrong quotient digit, like 4 initially (48>45), or subtraction mistake, such as 45-36=11. The algorithm ensures the remainder is always less than the divisor; if not, increase the quotient digit. This applies to distributing pencils, where 38 packs of 12 are made, with 1 pencil left over.
Marcus divides $7,245$ by $35$ and gets a quotient of $207$ . To check his work, he multiplies $207 × 35$ and gets $7,245$ . What can you conclude about his division?
Answer: His division is completely correct with no remainder needed When checking division problems, you need to verify that the multiplication actually works out correctly. Let's examine Marcus's work step by step. Marcus claims that $7,245 ÷ 35 = 207$ , and he checks this by multiplying $207 × 35$ . However, when you actually calculate $207 × 35$ , you get $7,235$ , not $7,245$ . You can verify this: $207 × 35 = 207 × (30 + 5) = 207 × 30 + 207 × 5 = 6,210 + 1,035 = 7,245$ . Wait, let me recalculate: $207 × 30 = 6,210$ and $207 × 5 = 1,035$ , so $6,210 + 1,035 = 7,245$ . Actually, Marcus's multiplication is correct. Let me check the division properly: $7,245 ÷ 35$ . Using long division or a calculator, $35 × 207 = 7,245$ exactly. So Marcus's work is actually completely correct. Looking at the answer choices: Choice A suggests his division is completely correct, which appears true. Choice B claims the quotient should be $270$ , but $270 × 35 = 9,450$ , which is way too large. Choice C suggests there should be a remainder, but since $207 × 35 = 7,245$ exactly, there's no remainder. Choice D claims that $207 × 35 = 7,235$ , but this is incorrect—the actual product is $7,245$ . Study tip: Always double-check your multiplication when verifying division. Calculate step-by-step rather than assuming, and remember that if division is exact, there will be no remainder.
Divide using the standard long division algorithm: $5724\div 18$ . Give the quotient and remainder.
Answer: $318\text{ R }0$ This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, repeat until complete, with quotient and remainder. For 5724 ÷ 18, divide 18 into 57 (goes 3 times, 18×3=54≤57<18×4=72), write 3, multiply 54, subtract 3, bring down 2 to make 32; 18 into 32 goes 1 time (18×1=18≤32<18×2=36), multiply 18, subtract 14, bring down 4 to make 144; 18 into 144 goes 8 times (18×8=144), multiply 144, subtract 0, so quotient 318 R0. Verify: 318×18=5724, exact match. The correct answer is 318 R0, as in choice A. Errors could be subtraction mistakes, like 57-54=4 instead of 3, or choosing too small a quotient digit. The remainder must be less than the divisor; here it's zero, meaning exact division. This algorithm is useful for equal sharing, like distributing items evenly.
A coach has 864 water bottles to pack equally into 9 bins. Using the standard long division algorithm, find $864\div 9$ .
Answer: 96 This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 864 ÷ 9, start by dividing 9 into 86 (9 goes 9 times since 9×9=81≤86<9×10=90), write 9 above, multiply 9×9=81, subtract 86-81=5, bring down 4 to make 54; then 9 into 54 goes 6 times (9×6=54), multiply 54, subtract 0, so quotient is 96 with no remainder. Verify by multiplying quotient by divisor: 96×9=864, which equals the dividend. The correct answer is 96, as choice A states. A common mistake is choosing a quotient digit too large, like 10 for the first step, leading to incorrect subtraction. Remember, at each step, determine how many times the divisor fits into the current number without exceeding it, and always check that the remainder is less than the divisor. In contexts like packing water bottles, this means each of the 9 bins gets 96 bottles exactly.
Which equation correctly verifies the division $3{,}409\div 27 = 126\ \text{R}7$ ?
Answer: $27\times 126 + 7 = 3{,}409$ This question tests understanding verification of division with remainder, where quotient × divisor + remainder should equal the dividend. For 3,409 ÷ 27 = 126 R7, the correct equation is 27×126 +7=3,409, as 126×27=3,402 and +7=3,409 matches. Other options like 27×126 -7=3,395 do not equal 3,409. This verification confirms the division is accurate. A common mistake is confusing the order, like multiplying remainder by quotient instead. Always use the formula quotient × divisor + remainder = dividend to check. This concept ensures reliability in division results, useful in various mathematical contexts.
A library received $1{,}848$ new bookmarks and wants to place them equally into $7$ bins. Divide $1{,}848\div 7$ using the standard long division algorithm. How many bookmarks go in each bin?
Answer: 264 This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 1,848 ÷ 7, divide 7 into 18, which goes 2 times (7×2=14), subtract 18-14=4, bring down 4 to make 44; 7 into 44 goes 6 times (7×6=42), subtract 44-42=2, bring down 8 to make 28; 7 into 28 goes 4 times (7×4=28), subtract 0, quotient 264. Verify: 264×7=1,848, matches. Each bin gets 264 bookmarks. A frequent mistake is incorrect quotient selection, like 3 for 18 (7×3=21>18). Always trial multiply to find the largest fitting digit. This context shows equal distribution of items into groups.
A science teacher pours 3,432 milliliters of solution equally into 16 cups. Using the standard long division algorithm, find $3432\div 16$ as a quotient with a remainder.
Answer: $214\text{ R }8$ This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, repeat until complete, with quotient and remainder. For 3432 ÷ 16, divide 16 into 34 (goes 2 times, 16×2=32≤34<16×3=48), write 2, multiply 32, subtract 34-32=2, bring down 3 to make 23; 16 into 23 goes 1 time (16×1=16≤23<16×2=32), multiply 16, subtract 7, bring down 2 to make 72; 16 into 72 goes 4 times (16×4=64≤72<16×5=80), multiply 64, subtract 8, so quotient 214 R8. Verify: 214×16 +8=3424+8=3432, correct. The correct answer is 214 R8, as in choice A. Common mistakes include misalignment of place values or forgetting to bring down digits, leading to wrong quotients like 215. Always trial multiply to choose the right quotient digit, ensuring the product doesn't exceed the current number. In pouring solution, this means each of 16 cups gets 214 ml, with 8 ml left over.
Divide using the standard long division algorithm: $4{,}536\div 12$ . What is the quotient?
Answer: $378$ This question tests dividing multi-digit numbers using the standard long division algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 4,536 ÷ 12, divide 12 into 45 (3 times since 12×3=36 ≤45 <12×4=48), write 3, multiply 36, subtract 9, bring down 3 to make 93; 12 into 93 goes 7 times (12×7=84), subtract 9, bring down 6 to make 96; 12 into 96 goes 8 times (12×8=96), subtract 0, so quotient 378 with no remainder. Verify: 378×12=4,536, which matches. The correct quotient is 378, as in option A. A mistake like 37 R8 might come from not bringing down properly or subtraction error. Always align quotient digits with place values. This example highlights repeating steps until all digits are processed.
Calculate using the standard long division algorithm: $7{,}245\div 9$ .
Answer: $805$ This question tests dividing multi-digit numbers using the standard long division algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 7,245 ÷ 9, divide 9 into 72 (8 times since 9×8=72), write 8, multiply 72, subtract 0, bring down 4 to make 4; 9 into 4 goes 0 times, write 0, bring down 5 to make 45; 9 into 45 goes 5 times (9×5=45), subtract 0, so quotient 805 with no remainder. Verify: 805×9=7,245, which matches. The correct quotient is 805, as in option A. Errors might include 815 if adding instead of subtracting correctly or misaligning places. Remember to include zeros in the quotient when a digit doesn't divide. This calculation demonstrates the algorithm's steps clearly.
A teacher has $864$ pencils to pack equally into $24$ supply boxes. Using the standard long division algorithm, how many pencils go in each box?
Answer: $36$ This question tests dividing multi-digit numbers using the standard long division algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 864 ÷ 24, start by dividing 24 into 86 (3 times since 24×3=72 ≤86 <24×4=96), write 3 above, multiply 24×3=72, subtract 86-72=14, bring down 4 to make 144; then 24 into 144 goes 6 times (24×6=144), multiply and subtract to get 0, so quotient is 36 with no remainder. Verify by multiplying: 36×24=864, which matches the dividend. The correct answer is 36 pencils per box, as chosen in option B. A common mistake might be choosing 34 if subtracting incorrectly, like 86-72=16 instead of 14, leading to a wrong quotient. Remember, at each step, choose the largest digit where the product is less than or equal to the current number, and always check that the remainder is less than the divisor. This context of packing pencils equally demonstrates fair distribution, and verification ensures accuracy.
A bakery uses $4,368$ pounds of flour equally over $24$ days. If they package the daily flour amount into bags that each hold $18$ pounds, how many bags do they need each day?
Answer: $10$ bags with $2$ pounds left over First, divide total flour by days: $4,368 ÷ 24 = 182$ pounds per day. Then divide daily amount by bag capacity: $182 ÷ 18 = 10$ remainder $2$ . So they need $10$ bags with $2$ pounds left over. Choice A uses incorrect daily amount ( $4,368 ÷ 26 = 168$ ). Choice C assumes $216$ pounds per day. Choice D uses $174$ pounds per day.
Divide using the standard long division algorithm: $2{,}835\div 15$ . What is the quotient?
Answer: $189$ This question tests dividing multi-digit numbers using the standard long division algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 2,835 ÷ 15, divide 15 into 28 (1 time since 15×1=15 ≤28 <15×2=30), write 1, multiply 15, subtract 13, bring down 3 to make 133; 15 into 133 goes 8 times (15×8=120), subtract 13, bring down 5 to make 135; 15 into 135 goes 9 times (15×9=135), subtract 0, so quotient 189 with no remainder. Verify: 189×15=2,835, which matches. The correct quotient is 189, as in option B. Mistakes like 180 R135 might occur from stopping early or misalignment. Ensure remainder is always less than divisor. This shows exact division in practice.
A classroom collected $864$ pencils to donate and wants to pack them equally into $24$ boxes. Divide $864\div 24$ using the standard long division algorithm. How many pencils go in each box?
Answer: 36 This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 864 ÷ 24, start by dividing 24 into 86, which goes 3 times since 24×3=72, subtract 86-72=14, bring down 4 to make 144; then 24 into 144 goes 6 times since 24×6=144, subtract 144-144=0, so the quotient is 36 with no remainder. To verify, multiply the quotient by the divisor: 36×24=864, which matches the dividend. The correct quotient is 36, meaning each box gets 36 pencils. A common mistake might be choosing a quotient digit too large, like 4 for the first step (24×4=96>86), leading to incorrect subtraction. Remember, at each step, determine how many times the divisor fits into the current number without exceeding it, then proceed with multiply, subtract, and bring down. This division context represents equally distributing pencils into boxes, ensuring fair sharing.
Divide using the standard long division algorithm: $3065\div 7$ . Give the quotient and remainder.
Answer: $437\text{ R }6$ This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, repeat until complete, with quotient and remainder. For 3065 ÷ 7, divide 7 into 30 (goes 4 times, 7×4=28≤30<7×5=35), write 4, multiply 28, subtract 2, bring down 6 to make 26; 7 into 26 goes 3 times (7×3=21≤26<7×4=28), multiply 21, subtract 5, bring down 5 to make 55; 7 into 55 goes 7 times (7×7=49≤55<7×8=56), multiply 49, subtract 6, so quotient 437 R6. Verify: 437×7 +6=3059+6=3065, matches. The correct answer is 437 R6, as in choice A. Common errors include subtraction mistakes or choosing a quotient digit too large, like 8 for 55 (56>55). Ensure each remainder is less than the divisor before proceeding. This helps in contexts like grouping items evenly.
Verify a division result by multiplication: If $4536\div 36 = 126$ , which check is correct?
Answer: $36\times 126 = 4536$ This question tests verifying division results by multiplication, a key step after using the standard algorithm to ensure accuracy. To check 4536 ÷ 36 =126, multiply 36×126: 36×100=3600, 36×20=720, 36×6=216, total 3600+720+216=4536, which equals the dividend, confirming it's correct. The proper verification is quotient × divisor = dividend (or +remainder if any), so choice A is right. Incorrect options might swap numbers or have arithmetic errors, like 4563 instead of 4536. Always perform this multiplication check after division to catch mistakes in the algorithm steps. Common division errors include wrong quotient digits or subtraction, which this verification reveals. This method applies to real-world scenarios like confirming equal shares.
Which equation correctly verifies the division $3{,}927\div 18 = 218\text{ R }3$ ?
Answer: $18\times 218 + 3 = 3{,}927$ This question tests understanding verification of multi-digit division using the standard algorithm, where dividend equals divisor times quotient plus remainder. To verify 3,927 ÷ 18 = 218 R3, check if 18×218 +3=3,927, which is the correct equation. Other options like subtracting the remainder or reversing operations don't match the formula. The correct choice is option B. A common mistake is thinking remainder is subtracted, but it's added back to reconstruct the dividend. Always use quotient×divisor + remainder = dividend for checks. This reinforces the importance of verification after division.
A store has $2{,}019$ stickers and packs them into $16$ identical packs. Divide $2{,}019\div 16$ using the standard long division algorithm. What is the quotient with remainder?
Answer: $126\ \text{R}3$ This question tests dividing multi-digit numbers using the standard algorithm: divide, multiply, subtract, bring down, and repeat until complete, resulting in a quotient and possibly a remainder. For 2,019 ÷ 16, divide 16 into 20, which goes 1 time (16×1=16), subtract 20-16=4, bring down 1 to make 41; 16 into 41 goes 2 times (16×2=32), subtract 41-32=9, bring down 9 to make 99; 16 into 99 goes 6 times (16×6=96), subtract 99-96=3, quotient 126 R3. Verify: 126×16 +3=2,016+3=2,019, matches. The correct answer is 126 R3, with 3 stickers left over. Mistakes often stem from wrong trial multiplication or forgetting to bring down. Ensure remainder < divisor. This represents packing items equally with possible leftovers.

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