Answer: No — the worker can only produce 81 pieces with 8 cm left over

Perform the division: $980 \div 12 = 81$ remainder $8$, since $81 \times 12 = 972$ and $980 - 972 = 8$. Interpret the remainder in a cutting context: When cutting exact lengths, you can only count complete pieces. The remaining 8 cm is too short to make another 12 cm piece, so only 81 pieces can be produced. Why distractors fail: Option B: $82 \times 12 = 984 > 980$, so 82 pieces is impossible. Option C: Uses incorrect remainder arithmetic. Option D: Rounding up is appropriate when every item must be accommodated (e.g., buses), but here you cannot create a piece longer than the available ribbon.

Answer: 7 hours

Find the combined rate: 4 volunteers each planting 125 per hour gives a combined rate of $4 \times 125 = 500$ seedlings per hour. Divide total by combined rate: $3{,}500 \div 500 = 7$ hours. Why distractors fail: Option A (5): Would require a combined rate of 700 per hour. Option C (8): $8 \times 500 = 4{,}000 \neq 3{,}500$. Option D (28): This is $3{,}500 \div 125$, i.e., the time for one volunteer alone.

Answer: 6 dollars per km

Subtract the flat fee: Amount attributed to distance = $95 - 11 = 84$ dollars. Divide to find the per-km rate: $84 \div 14 = 6$ dollars per km. Verify: $11 + 6 \times 14 = 11 + 84 = 95$ ✓ Why distractors fail: Option A: $11 + 5 \times 14 = 81 \neq 95$. Option C: $11 + 7 \times 14 = 109 \neq 95$. Option D: This results from $95 \div 14$ without removing the flat fee first.

Answer: 6.50 dollars per kg

Set up the unit price calculation: Unit price = total cost ÷ quantity = $52 \div 8$. Compute: $52 \div 8 = 6.5$. The unit price is 6.50 dollars per kg. Why distractors fail: Option A: $6.00 \times 8 = 48 \neq 52$. Option B: $6.25 \times 8 = 50 \neq 52$. Option D: $7.00 \times 8 = 56 \neq 52$.

Answer: 168 kg of each

Set up the equation: Let $k$ = kilograms of each fruit. Total cost = $3k + 2k = 5k$. Budget constraint: $5k = 840$. Solve for k: $k = 840 \div 5 = 168$. They can buy 168 kg of apples and 168 kg of bananas. Verify: $168 \times 3 + 168 \times 2 = 504 + 336 = 840$ ✓ Why distractors fail: Option A (140): $140 \times 5 = 700 \neq 840$. Option C (210): $210 \times 5 = 1{,}050 > 840$. Option D (280): $280 \times 5 = 1{,}400$, far exceeding the budget.

Answer: 52 km/h

Identify the operation: Speed = total distance ÷ total time = $364 \div 7$. Compute the quotient: $364 \div 7 = 52$. The speed is 52 km/h. Why distractors fail: Option A (48): $48 \times 7 = 336 \neq 364$. Option B (50): $50 \times 7 = 350 \neq 364$. Option D (54): $54 \times 7 = 378 \neq 364$.

Answer: The remainder is wrong; $945 - 30 \times 32 = 945 - 960$, which is negative, so the quotient is too large

Check the student's work: $30 \times 32 = 960$. But $960 > 945$, so a quotient of 30 is too large. The product exceeds the dividend. Find the correct answer: $29 \times 32 = 928$. Remainder = $945 - 928 = 17$. The correct answer is 29 full boxes with 17 postcards left over. Why distractors fail: Option A: 29 is actually correct, but the reasoning given ('rounded up incorrectly') is vague and does not identify the core issue. Option B correctly identifies that the product exceeds the dividend, making the quotient too large. Option C: Rounding up to 31 would be appropriate for the number of boxes needed, but the question asks about the student's long division error, not context interpretation. Option D: The answer is demonstrably incorrect.

Answer: Brand A at 2.40 dollars per litre

Compute unit price for Brand A: $3.60 \div 1.5 = 2.40$ dollars per litre. Compute unit price for Brand B: $5.00 \div 2 = 2.50$ dollars per litre. Compare and conclude: Brand A (2.40 dollars/litre) is cheaper per litre than Brand B (2.50 dollars/litre). Why distractors fail: Option B: Correctly states Brand B's unit price but presents it as the lower one. Option C: The two unit prices differ. Option D: $2.00$ per litre would require $2.00 \times 2 = 4.00$ dollars, not 5.00.

Answer: Round the quotient up to the next whole number so no passengers are left behind

Perform the division: $215 \div 40 = 5$ remainder $15$. Five buses hold $5 \times 40 = 200$ passengers, leaving 15 passengers. Interpret the remainder in context: Because every passenger must have a seat, you need an additional bus for the remaining 15 passengers, giving 6 buses total. Why distractors fail: Option A: Ignoring the remainder leaves 15 passengers without transport. Option B: Reporting a fraction does not answer how many whole buses are needed. Option D: Rounding down also strands 15 passengers.

Answer: 37

Perform the division: $875 \div 24 = 36$ remainder $11$, because $36 \times 24 = 864$ and $875 - 864 = 11$. Interpret the remainder: Since there are 11 leftover items that still need to be shipped, an additional crate is required. Total crates = $36 + 1 = 37$. Why distractors fail: Option A (36): This would leave 11 items unshipped. Option C (35): $35 \times 24 = 840$, leaving 35 items unshipped. Option D (38): One more crate than necessary.

Answer: 26 books each, 2 left over

Set up the long division: Compute $1{,}250 \div 48$. Estimate: $48 \times 26 = 1{,}248$. Find quotient and remainder: $1{,}250 - 1{,}248 = 2$. So the quotient is 26 and the remainder is 2. Verify: Check: $26 \times 48 + 2 = 1{,}248 + 2 = 1{,}250$. ✓ Why distractors fail: Option A: $25 \times 48 = 1{,}200$, remainder would be 50, not matching the computation correctly — though the arithmetic is consistent, 26 is a better quotient because $48 \times 26 = 1{,}248 < 1{,}250$. Option C: There is a remainder of 2. Option D: $25 \times 48 = 1{,}200 \neq 1{,}250$.

Answer: 25

Perform the division: $150 \div 6 = 25$ with no remainder. Interpret the result: Since 150 divides evenly by 6, exactly 25 boxes are needed with no leftovers. Why distractors fail: Option A (24): $24 \times 6 = 144$, leaving 6 bottles unpacked. Option C (26): one extra unnecessary box. Option D (30): $30 \times 6 = 180$, far too many boxes.

Answer: 12

Set up the division: We need to divide 84 stickers equally among 7 students: $84 \div 7$. Compute the quotient: $84 \div 7 = 12$. Each student receives 12 stickers. Why distractors fail: Option A (11): $11 \times 7 = 77$, not 84. Option C (13): $13 \times 7 = 91$, not 84. Option D (14): $14 \times 7 = 98$, not 84.

Answer: Each pays 47 dollars; the remaining 2 dollars is split further or covered by volunteers

Perform the division: $237 \div 5 = 47$ remainder $2$. Each person pays 47 dollars, but $5 \times 47 = 235$, which is 2 dollars short. Interpret the remainder in a cost-sharing context: The remaining 2 dollars must still be paid. Reasonable approaches include having 2 of the 5 friends each pay 1 extra dollar, or handling it in another agreed-upon way. The total collected must equal 237 dollars. Why distractors fail: Option A: Ignoring 2 dollars means the gift is not fully paid for. Option B: If each pays 48 dollars the total is 240, overpaying by 3, which is inefficient and inaccurate. Option D: There is no basis for assuming the store would give a discount.

Answer: No — $2{,}340 \div 36 = 65$ exactly, so 65 crates are needed, not 64

Perform the division: $2{,}340 \div 36 = 65$ with remainder 0, because $65 \times 36 = 2{,}340$. Evaluate the claim: Since 65 crates are needed with no remainder, the worker's claim of 64 crates is incorrect — 64 crates would only hold $64 \times 36 = 2{,}304$ apples, leaving 36 unpacked. Why distractors fail: Option A: $64 \times 36 = 2{,}304 \neq 2{,}340$. Option B: States 65 crates are needed (correct number) but incorrectly claims there is a remainder of 36. Option C: Incorrectly suggests the last crate is not full; the division is exact.

Answer: Because some situations require rounding up, some require rounding down, and some require reporting the remainder itself, depending on what is being divided and why

Core principle: In real-world problems the remainder carries meaning. Whether you need an extra container (round up), can only use complete groups (round down), or must report what is left over depends entirely on the scenario. Why the correct answer works: Option B captures the full range of remainder interpretations: rounding up (e.g., needing an extra vehicle), rounding down (e.g., how many complete sets can be made), or stating the remainder (e.g., leftover items to redistribute). Why distractors fail: Option A: The size of the divisor is irrelevant to how remainders are interpreted. Option C: Decimal expression is one possibility but not always appropriate (e.g., you cannot have 0.5 of a bus). Option D: Remainders are not always discarded — sometimes they are the most important part of the answer.

Answer: The number of complete bread rolls the baker can make

Identify dividend and divisor meanings: The dividend (500 g) is the total resource. The divisor (65 g) is the amount needed per item. The quotient tells how many complete items can be produced. Perform the division: $500 \div 65 = 7$ remainder $45$. The baker can make 7 complete rolls with 45 g left over. Why distractors fail: Option A describes the remainder, not the quotient. Option B: Total flour used would be $7 \times 65 = 455$ g, a separate calculation. Option D: The grams per roll is the divisor (given), not the quotient.

Answer: The coordinator should have ordered 26 buses to accommodate the 29 remaining students

Verify the division: $45 \times 25 = 1{,}125$ and $1{,}154 - 1{,}125 = 29$. The arithmetic $1{,}154 \div 45 = 25$ R $29$ is correct. Identify the contextual error: The 29 remaining students still need a bus. In transportation problems, the remainder means you must round up: $25 + 1 = 26$ buses. Why distractors fail: Option A: The division arithmetic is actually correct. Option C: Ignoring the remainder leaves 29 students without transportation. Option D: Fractional buses do not exist in practice; the answer must be a whole number rounded up.