Answer: $0.45 \div 0.0002$

Count decimal places in each divisor: $0.07$ has 2 places (need $\times 100$). $0.003$ has 3 places (need $\times 1{,}000$). $0.0002$ has 4 places (need $\times 10{,}000$). $0.9$ has 1 place (need $\times 10$). Why the correct answer works: Only $0.0002$ requires multiplying by $10{,}000$ to become the whole number $2$. Why distractors fail: Option A needs $\times 100$. Option B needs $\times 1{,}000$. Option D needs $\times 10$.

Answer: It creates an equivalent division problem with a whole-number divisor, which is easier to compute

Identify the purpose: The goal is to transform the problem so the divisor is a whole number, since long division with a whole-number divisor follows a straightforward algorithm. Why the correct answer works: Multiplying both by the same power of 10 produces an equivalent quotient (by the Equivalent Division Property) and removes the decimal from the divisor. Why distractors fail: Option A is wrong because the numbers generally get larger, not smaller. Option C is wrong because the quotient may still have decimals. Option D is wrong because the operation remains division.

Answer: Multiplying both the divisor and dividend by the same number does not change the quotient

Recall the property: The Equivalent Division Property states that $a \div b = (a \times n) \div (b \times n)$ for any nonzero $n$. Why the correct answer works: Option B correctly restates this principle: multiplying both numbers by the same factor leaves the quotient unchanged. Why distractors fail: Option A only multiplies the dividend, which would change the quotient. Option C uses addition, which does not preserve the quotient. Option D uses different numbers for each, which also changes the quotient.

Answer: Multiply $70 \times 0.12$ and check if the product equals $8.4$

Apply the inverse operation: The standard check for division is: quotient $\times$ divisor = dividend. Here, $70 \times 0.12 = 8.4$ ✓, confirming the answer is correct. Why the correct answer works: Option A applies the correct inverse-operation check. Since multiplication undoes division, if the product matches the dividend, the quotient is verified. Why distractors fail: Option B divides the dividend by the quotient — while this would also yield $0.12$, it is a less standard and less direct verification method. Option C is valid in theory but impractical with 70 iterations. Option D uses repeated subtraction but would require 70 steps, making it inefficient.

Answer: $0.48 \div 0.006$

Reverse-engineer the problem: If multiplying by $1{,}000$ (because the divisor has 3 decimal places) gives $480 \div 6$, then the original dividend is $480 \div 1000 = 0.48$ and the original divisor is $6 \div 1000 = 0.006$. Verify all conditions: $0.006$ has 3 decimal places ✓. $0.48 \times 1000 = 480$, $0.006 \times 1000 = 6$ → $480 \div 6 = 80$ ✓. Why distractors fail: Option B: $0.06$ has only 2 decimal places. Option C: $0.048 \times 1000 = 48$, giving $48 \div 6 = 8$, not 80. Option D: $0.6$ has only 1 decimal place.

Answer: No, the student's calculation is correct; dividing by a number less than 1 produces a quotient larger than the dividend

Check the computation: $0.5 \div 0.2 = 5 \div 2 = 2.5$. The calculation is correct. Address the misconception: Division by a number between 0 and 1 yields a quotient larger than the dividend. This is analogous to asking "how many groups of $0.2$ fit in $0.5$?"—the answer is 2.5 groups. Why distractors fail: Option A perpetuates the misconception. Option C gives an incorrect value. Option D confuses division with multiplication.

Answer: $345 \div 15$

Determine the required scaling: $0.15$ has 2 decimal places, so multiply both by 100. Apply the scaling: $3.45 \times 100 = 345$ and $0.15 \times 100 = 15$. So $345 \div 15 = 23$ is equivalent. Why distractors fail: Option A scales the dividend by only 10 ($34.5$) but the divisor by 100 ($15$), giving $34.5 \div 15 = 2.3$. Option C scales the dividend by 1000 but divisor by 100, giving $3450 \div 15 = 230$. Option D keeps the divisor as a decimal.

Answer: Multiply both the divisor and the dividend by 100 to make the divisor a whole number

Identify the key principle: When dividing decimals by decimals, the standard strategy is to first make the divisor a whole number by multiplying both numbers by the same power of 10. Why the correct answer works: The divisor $0.12$ has 2 decimal places, so multiplying both by $100$ gives $456 \div 12$, which is an equivalent division that is easier to compute. Why distractors fail: Option A is wrong because moving the decimal only in the dividend changes the quotient. Option C is wrong because rounding the divisor changes the problem and introduces error. Option D is wrong because placing the decimal in the quotient is a later step, not the first.

Answer: $912 \div 4$

Count decimal places in the divisor: $0.04$ has 2 decimal places, so multiply both by 100. Perform the scaling: $9.12 \times 100 = 912$ and $0.04 \times 100 = 4$. So $9.12 \div 0.04 = 912 \div 4$. Why distractors fail: Option A only multiplies by 10 ($91.2 \div 0.4$, and the divisor is still not whole either — actually $91.2 \div 4$ would mean multiplying divisor by 100 but dividend by only 10, giving the wrong quotient). Option C multiplies by 1000 for the dividend but 100 for the divisor. Option D only multiplies by 10, leaving the divisor as $0.4$.

Answer: Directly above the decimal point in the adjusted dividend $24.6$

Understand the adjustment: Multiplying both by 10 converts $2.46 \div 0.3$ to $24.6 \div 3$. Why the correct answer works: The decimal in the quotient is aligned directly above the decimal point in the adjusted dividend $24.6$, ensuring correct place value. Why distractors fail: Option A uses the original dividend's decimal, which has shifted. Option C arbitrarily places the decimal at the end. Option D is wrong because $24.6 \div 3 = 8.2$, which does have a decimal.

Answer: $1{,}000$

Count the decimal places in the divisor: The divisor $0.005$ has 3 decimal places. Determine the power of 10: To eliminate 3 decimal places, multiply by $10^3 = 1{,}000$. This converts $0.005$ to $5$ and $7.35$ to $7{,}350$. Why distractors fail: Option A ($10$) would give $0.05$, still a decimal. Option B ($100$) would give $0.5$, still a decimal. Option D ($10{,}000$) is more than needed and while it would work mathematically, $1{,}000$ is the standard minimal power of 10 to make the divisor whole.

Answer: The method is correct and the answer is 6 bottles

Verify the method: The divisor $0.125$ has 3 decimal places, so multiplying by $1{,}000$ is correct. $0.75 \times 1000 = 750$ and $0.125 \times 1000 = 125$. Verify the arithmetic: $750 \div 125 = 6$. The cook needs exactly 6 bottles. Why distractors fail: Option A gives the wrong answer despite approving the method. Option C uses 100, which leaves the divisor as $12.5$ (still a decimal). Option D violates the Equivalent Division Property by scaling only one number.

Answer: 63

Make the divisor whole: The divisor $0.25$ has 2 decimal places. Multiply both by 100: $1575 \div 25$. Perform the division: $1575 \div 25 = 63$. Why distractors fail: Option A ($6.3$) results from multiplying only by 10. Option C ($630$) results from multiplying by an extra power of 10. Option D ($0.63$) reflects a major decimal placement error.

Answer: 25

Make the divisor whole: The divisor $0.005$ has 3 decimal places. Multiply both by $1{,}000$: $125 \div 5$. Perform the division: $125 \div 5 = 25$. Why distractors fail: Option A ($2.5$) results from multiplying by only 100. Option C ($250$) results from multiplying by 10000. Option D ($0.25$) reflects a serious decimal placement error.

Answer: It will be 10 times too small

Analyze the unequal scaling: The student computed $\frac{0.36 \times 100}{0.006 \times 1000} = \frac{36}{6} = 6$. The correct computation: $\frac{0.36 \times 1000}{0.006 \times 1000} = \frac{360}{6} = 60$. Quantify the error: The student's answer (6) is $\frac{1}{10}$ of the correct answer (60), so the quotient is 10 times too small. Why distractors fail: Option A is the opposite direction. Option C is wrong because the scaling was unequal. Option D would require multiplying the dividend by 10 and divisor by 1000.

Answer: 570

Set up the division: $0.684 \div 0.0012$. The divisor has 4 decimal places, so multiply both by $10{,}000$. Perform the adjusted division: $6840 \div 12 = 570$. Why distractors fail: Option A (57) results from multiplying by only 1000. Option C (5700) results from multiplying by 100000. Option D (5.7) reflects a major decimal error.

Answer: The decimal should be after the 1, making the answer $1.2$, which is correct

Verify the adjustment: $0.012$ has 3 decimal places, so multiply both by $1{,}000$: $0.0144 \times 1000 = 14.4$ and $0.012 \times 1000 = 12$. Perform the division with decimal placement: $14.4 \div 12$: The decimal in the quotient aligns above the decimal in $14.4$. $12 \div 12 = 1$ (ones place), then bring down $4$: $24 \div 12 = 2$ (tenths place). Quotient = $1.2$. Why distractors fail: Options B and D give $12$, which would mean $12 \times 12 = 144 \neq 14.4$. Option C gives $0.12$, which would mean $0.12 \times 12 = 1.44 \neq 14.4$.

Answer: 4

Count the decimal places: In $0.0045$, starting after the decimal point, there are 4 digits: 0, 0, 4, 5. Thus there are 4 decimal places. Why the correct answer works: The number of decimal places is the total count of digits to the right of the decimal point, regardless of whether some are zeros. Why distractors fail: Option A (2) only counts the significant digits 4 and 5. Option B (3) miscounts. Option D (5) counts the digit before the decimal point.

Answer: 9

Make the divisor whole: Multiply both by 10: $3.6 \times 10 = 36$ and $0.4 \times 10 = 4$. Perform the division: $36 \div 4 = 9$. Why distractors fail: Option A ($0.9$) results from dividing only the dividend by 10 extra. Option C ($90$) results from multiplying by 100 instead of 10 for just the dividend. Option D ($0.09$) misplaces the decimal point significantly.

Answer: Both students, because both multiplied divisor and dividend by the same power of 10

Analyze both approaches: The first student multiplied by 100: $2.4 \times 100 = 240$, $0.03 \times 100 = 3$. The second multiplied by 1000: $2.4 \times 1000 = 2400$, $0.03 \times 1000 = 30$. Apply the Equivalent Division Property: Both are valid because the Equivalent Division Property applies for any nonzero multiplier. Both yield 80. Why distractors fail: Option A wrongly insists the divisor must be single-digit. Option B wrongly claims larger numbers are more accurate. Option D incorrectly calculates the quotient.

Answer: They incorrectly adjusted the quotient; after converting to $48 \div 6 = 8$, no further decimal adjustment is needed

Identify the student's process: The student correctly multiplied both by 10 to get $48 \div 6 = 8$. However, they then unnecessarily moved the decimal in the quotient. Apply the Equivalent Division Property: Since both numbers were scaled equally, the quotient $8$ is already correct. $4.8 \div 0.6 = 8$, not $0.8$. Why distractors fail: Option A suggests further decimal movement, compounding the error. Option C ignores the need to eliminate the decimal divisor. Option D suggests using 100, which is unnecessary since 10 suffices.

Answer: All three students

Check each student's scaling: Student A: both $\times 100$ → $720 \div 9 = 80$. Student B: both $\times 10$ → $72 \div 0.9$. To evaluate: $72 \div 0.9 = 720 \div 9 = 80$. Student C: both $\times 1000$ → $7200 \div 90 = 80$. Why the correct answer works: All three multiplied both numbers by the same factor, so all get 80 by the Equivalent Division Property. Why distractors fail: Options A, B, and D all exclude at least one valid approach. The Equivalent Division Property applies to any nonzero scaling factor.

Answer: They have the same quotient because both the divisor and dividend were multiplied by the same power of 10

Apply the Equivalent Division Property: $6.4 \div 0.08 = (6.4 \times 100) \div (0.08 \times 100) = 640 \div 8$. Since both were multiplied by 100, the quotient is the same. Why the correct answer works: Both expressions equal $80$, confirming they are equivalent divisions. Why distractors fail: Option A ignores the Equivalent Division Property. Option C incorrectly assumes the quotient is scaled. Option D is factually wrong—neither has a remainder.

Answer: 8.1

Make the divisor whole: The divisor $0.08$ has 2 decimal places. Multiply both by 100: $64.8 \div 8$. Perform the division: $64.8 \div 8 = 8.1$. Why distractors fail: Option A ($81$) misplaces the decimal one position to the right. Option B ($0.81$) misplaces it one position to the left. Option D ($0.081$) is off by two decimal places.

Answer: Multiplied by 100; the conversion is correct

Determine the power of 10: $0.07$ has 2 decimal places, so the student should multiply both by $100$. $0.07 \times 100 = 7$ and $5.6 \times 100 = 560$. Verify the conversion: $560 \div 7 = 80$, which matches $5.6 \div 0.07 = 80$. The conversion is correct. Why distractors fail: Option A says multiplied by 10, but $0.07 \times 10 = 0.7$, not 7. Option C claims the dividend should be 56, but $5.6 \times 100 = 560$. Option D uses 1000, which is too large.