This problem tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. The four operations with rationals include addition/subtraction using sign rules where same signs add magnitudes and different signs subtract while taking the larger's sign, and subtraction as adding the inverse like p - q = p + (-q), while multiplication/division follow sign rules where same signs give positive and different signs give negative; in multi-step problems, track values through the sequence such as start 0, add 12.5 to 12.5, subtract 18.75 to -6.25, then add (2/3)×9=6 to -0.25 m, with negatives for below launch point. For example, in a profit/loss average: Monday +$120, Tuesday -$45, Wednesday +$80, calculate (120 + (-45) + 80) ÷ 3 = (120 - 45 + 80) ÷ 3 = 155 ÷ 3 ≈ $51.67 average daily profit; or temperature: start -12°C, rise 8° (add): -12 + 8 = -4°C, then drop half the rise: -4 - 4 = -8°C; or recipe: 2/3 cup for 4 servings, make 6 (multiply by 6/4 = 1.5): (2/3) × (3/2) = 1 cup. To solve correctly, 0 + 12 1/2 =12.5, 12.5 -18.75= -6.25, -6.25 + (2/3 9)= -6.25 +6= -0.25 m. A common error is sign error like 12.5 -18.75 as 6.25 positive, or fraction wrong 2/39 as 3 or 6 wrong to 5.4, or order wrong adding before subtracting, or arithmetic like -6.25+6 as -0.25 wrong to 0.25 ignoring sign. To solve these, (1) identify operations: add, subtract, multiply then add, (2) sequence: after each move, (3) apply sign rules for subtraction leading to negative, (4) convert mixed 12 1/2 to 12.5, (5) track: +12.5=12.5, -18.75=-6.25, +6=-0.25, (6) interpret as -0.25 m below launch. Context 'rises, drops, rises again by 2/3 of 9 m' suggests add, subtract, multiply then add, with negative meaning below zero.

Tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. Four operations with rationals: addition/subtraction (sign rules: same signs add magnitudes, different signs subtract, take larger's sign; subtraction as adding inverse: p-q=p+(-q)), multiplication/division (sign rules: same signs give positive, different signs give negative). Multi-step: track values through sequence (start $\frac{3}{4}$ L, subtract $0.2$ L→$\frac{11}{20}$ L, add $\frac{1}{3}$ L→$\frac{53}{60}$ L total, divide by 2→$\frac{53}{120}$ L per cup). Context: liquid measurements requiring fraction-decimal conversions and equal division. Correct solution: $\frac{3}{4} - 0.2 = \frac{3}{4} - \frac{1}{5} = \frac{15}{20} - \frac{4}{20} = \frac{11}{20}$ L, then $\frac{11}{20} + \frac{1}{3} = \frac{33}{60} + \frac{20}{60} = \frac{53}{60}$ L, finally $\frac{53}{60} ÷ 2 = \frac{53}{60} × \frac{1}{2} = \frac{53}{120}$ L per cup. Error like forgetting to divide by 2 (answer $\frac{53}{60}$), converting 0.2 wrong (as $\frac{2}{10}$ not simplified to $\frac{1}{5}$), or adding before subtracting ($\frac{3}{4}+\frac{1}{3}-0.2$). Solving: (1) identify all operations needed (subtract amount drunk, add refill amount, divide total by 2), (2) sequence operations (subtract→add→divide), (3) apply operation rules (convert 0.2 to fraction $\frac{1}{5}$ for easier calculation), (4) convert forms if needed (all to fractions with common denominators), (5) track through steps ($\frac{3}{4}→\frac{11}{20}→\frac{53}{60}→\frac{53}{120}$), (6) interpret ($\frac{53}{120}$ L is amount in each of the 2 cups).

This problem tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. The four operations with rationals include addition/subtraction using sign rules where same signs add magnitudes and different signs subtract while taking the larger's sign, and subtraction as adding the inverse like $p - q = p + (-q)$, while multiplication/division follow sign rules where same signs give positive and different signs give negative; in multi-step problems, track values through the sequence such as sum $12.5 -9 +3.5 -4.5 = (12.5 + 3.5) + (-9 -4.5) =16 -13.5=2.5$, then divide by 4 $=0.625$ average, with negatives for score decreases. For example, in a profit/loss average: Monday $+120$, Tuesday $-45$, Wednesday $+80$, calculate $(120 + (-45) + 80) ÷ 3 = (120 - 45 + 80) ÷ 3 = 155 ÷ 3 ≈ $51.67 average daily profit; or temperature: start $-12°C$, rise $8°$ (add): $-12 + 8 = -4°C$, then drop half the rise: $-4 - 4 = -8°C$; or recipe: $2/3$ cup for 4 servings, make 6 (multiply by $6/4 = 1.5$): $(2/3) × (3/2) = 1$ cup. To solve correctly, sum $12.5 + (-9) + \frac{7}{2} + (-4.5) =12.5 -9 +3.5 -4.5$, $12.5+3.5=16$, $-9-4.5=-13.5$, $16-13.5=2.5$, $2.5 / 4 =0.625$. A common error is operation order wrong like dividing each before summing, sign error like treating $-9$ as $+9$ summing to $29.5/4=7.375$, wrong operation, fraction wrong $\frac{7}{2}$ as 3 or 4, treating negative as positive, or arithmetic error like $2.5/4=0.65$ or $0.6$. To solve these, (1) identify operations: add all changes then divide by 4, (2) sequence as sum first then divide, (3) apply sign rules for addition with positives/negatives, (4) convert $\frac{7}{2}$ to 3.5, (5) track sum: $12.5-9=3.5$, $+3.5=7$, $-4.5=2.5$, $/4=0.625$, (6) interpret as average change positive meaning net gain per round. Operation priority in sum before divide, context 'average' suggests sum then divide, mistakes in sign errors or forgetting to include all terms.

This question tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. Four operations with rationals: addition/subtraction (sign rules: same signs add magnitudes, different signs subtract, take larger's sign; subtraction as adding inverse: p-q=p+(-q)), multiplication/division (sign rules: same signs give positive, different signs give negative). Multi-step: track values through sequence (start $50, subtract $18.50→$31.50, add $30→$61.50, divide by 3→$20.50 each). Context: negative numbers for debt/loss/below zero/descent (meaningful negatives). For example, profit/loss average: Monday +$120, Tuesday -$45, Wednesday +$80, calculate (120+(-45)+80)÷3=(120-45+80)÷3=155÷3≈$51.67 average daily profit; or temperature: start -12°C, rise 8° (add): -12+8=-4°C, then drop half the rise: -4-4=-8°C; or recipe: 2/3 cup for 4 servings, make 6 (multiply by 6/4=1.5): (2/3)×(3/2)=1 cup. To solve, $18.00 - $4.80 = $13.20, then $13.20 + $7.50 = $20.70, then donate (1/3)×$20.70 = $6.90, so $20.70 - $6.90 = $13.80. A common error is calculating the donation on the wrong balance, like after buying but before earning: (1/3)×$13.20 ≈ $4.40, $13.20 - $4.40 = $8.80 (not an option), or decimal arithmetic error like 20.70 - 6.90 as 14.70. Solving: (1) identify all operations needed (read problem: add, subtract, multiply, divide?), (2) sequence operations (order matters: (a+b)÷c≠a+(b÷c)), (3) apply operation rules (sign rules for all four: same→positive quotient/product, different→negative), (4) convert forms if needed (2/3 and 1.5 to common form), (5) track through steps (running value after each operation), (6) interpret (result in context: -$5 means owes $5, -8°C means below zero). Operation priority: handle within parentheses or natural groups first, then multiply/divide, then add/subtract (or left-to-right if equal precedence). Context clues: "share" suggests divide, "total" suggests add/multiply, "difference" suggests subtract, "of" with fraction suggests multiply. Mistakes: operation order, sign errors (most common across all four operations), arithmetic with decimals/fractions, context interpretation (negative meaning).

Tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. Four operations with rationals: addition/subtraction (sign rules: same signs add magnitudes, different signs subtract, take larger's sign; subtraction as adding inverse: p-q=p+(-q)), multiplication/division (sign rules: same signs give positive, different signs give negative). Multi-step: track values through sequence (start $25, subtract $6.75→$18.25, add $4.50→$22.75, multiply by 1/5 for donation→$4.55 donated, subtract from $22.75→$18.20 remaining). Context: money transactions with spending (subtract), receiving (add), and fractional donation. Correct solution: $25.00 - $6.75 = $18.25, then $18.25 + $4.50 = $22.75, then 1/5 × $22.75 = $4.55 donation, finally $22.75 - $4.55 = $18.20. Error like calculating 1/5 of original $25 instead of current amount ($5 donation→$17.75 final), or keeping 1/5 instead of donating 1/5 (keeps $4.55→wrong), or arithmetic error in subtraction ($22.75-$4.55=$18.75). Solving: (1) identify all operations needed (subtract sandwich cost, add repayment, find 1/5 of current total, subtract donation), (2) sequence operations (must calculate current balance before finding 1/5), (3) apply operation rules (subtraction for spending, addition for receiving), (4) convert forms if needed (1/5 as decimal 0.2 or keep as fraction), (5) track through steps ($25→$18.25→$22.75→donation $4.55→$18.20), (6) interpret (final $18.20 is money remaining after all transactions).

This question tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. Four operations with rationals: addition/subtraction (sign rules: same signs add magnitudes, different signs subtract, take larger's sign; subtraction as adding inverse: p-q=p+(-q)), multiplication/division (sign rules: same signs give positive, different signs give negative). Multi-step: track values through sequence (start $50, subtract $18.50→$31.50, add $30→$61.50, divide by 3→$20.50 each). Context: negative numbers for debt/loss/below zero/descent (meaningful negatives). For example, profit/loss average: Monday +$120, Tuesday -$45, Wednesday +$80, calculate (120+(-45)+80)÷3=(120-45+80)÷3=155÷3≈$51.67 average daily profit; or temperature: start -12°C, rise 8° (add): -12+8=-4°C, then drop half the rise: -4-4=-8°C; or recipe: 2/3 cup for 4 servings, make 6 (multiply by 6/4=1.5): (2/3)×(3/2)=1 cup. To solve, $18.00 - $4.80 = $13.20, then $13.20 + $7.50 = $20.70, then donate (1/3)×$20.70 = $6.90, so $20.70 - $6.90 = $13.80. A common error is calculating the donation on the wrong balance, like after buying but before earning: (1/3)×$13.20 ≈ $4.40, $13.20 - $4.40 = $8.80 (not an option), or decimal arithmetic error like 20.70 - 6.90 as 14.70. Solving: (1) identify all operations needed (read problem: add, subtract, multiply, divide?), (2) sequence operations (order matters: (a+b)÷c≠a+(b÷c)), (3) apply operation rules (sign rules for all four: same→positive quotient/product, different→negative), (4) convert forms if needed (2/3 and 1.5 to common form), (5) track through steps (running value after each operation), (6) interpret (result in context: -$5 means owes $5, -8°C means below zero). Operation priority: handle within parentheses or natural groups first, then multiply/divide, then add/subtract (or left-to-right if equal precedence). Context clues: "share" suggests divide, "total" suggests add/multiply, "difference" suggests subtract, "of" with fraction suggests multiply. Mistakes: operation order, sign errors (most common across all four operations), arithmetic with decimals/fractions, context interpretation (negative meaning).

This problem tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. The four operations with rationals include addition/subtraction using sign rules where same signs add magnitudes and different signs subtract while taking the larger's sign, and subtraction as adding the inverse like p - q = p + (-q), while multiplication/division follow sign rules where same signs give positive and different signs give negative; in multi-step problems, track values through the sequence such as original 3/4 cup for 6, half recipe (3/4)×(1/2)=3/8, then 3 batches 3/8×3=9/8 cups, no negatives here. For example, in a profit/loss average: Monday +$120, Tuesday -$45, Wednesday +$80, calculate (120 + (-45) + 80) ÷ 3 = (120 - 45 + 80) ÷ 3 = 155 ÷ 3 ≈ $51.67 average daily profit; or temperature: start -12°C, rise 8° (add): -12 + 8 = -4°C, then drop half the rise: -4 - 4 = -8°C; or recipe: 2/3 cup for 4 servings, make 6 (multiply by 6/4 = 1.5): (2/3) × (3/2) = 1 cup. To solve correctly, (3/4) × (1/2) = 3/8 for half recipe, then 3/8 × 3 = 9/8 cups. A common error is wrong operation like dividing instead of multiplying for batches, fraction operation wrong like (3/4 × 1/2) as 3/2 or 3/8 wrong to 3/6, or doing 1/2 of recipe then 3 batches as (3/4)/2 ×3 = (3/8)×3=9/8 same, but perhaps misreading as 1/2 the recipe and then 3 times original or something leading to 9/4. To solve these, (1) identify operations: multiply by 1/2, then multiply by 3, (2) sequence correctly: amount for half, then times 3, (3) apply fraction multiplication rules, (4) keep in fractions, (5) track steps: 3/4 × 1/2 = 3/8, 3/8 × 3 = 9/8, (6) interpret as total yogurt needed. Context clues like 'make 1/2 of the recipe, then make 3 batches of that smaller amount' suggests multiply by 1/2 then by 3, mistakes in fraction multiplication or order.

This problem tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. The four operations with rationals include addition/subtraction using sign rules where same signs add magnitudes and different signs subtract while taking the larger's sign, and subtraction as adding the inverse like p - q = p + (-q), while multiplication/division follow sign rules where same signs give positive and different signs give negative; in multi-step problems, track values through the sequence such as starting at -6.5°C, adding 4 to get -2.5°C, then multiplying 4 by 3/2=6, then subtracting 6 to get -8.5°C, with negatives meaningful for below zero temperatures. For example, in a profit/loss average: Monday +$120, Tuesday -$45, Wednesday +$80, calculate (120 + (-45) + 80) ÷ 3 = (120 - 45 + 80) ÷ 3 = 155 ÷ 3 ≈ $51.67 average daily profit; or temperature: start -12°C, rise 8° (add): -12 + 8 = -4°C, then drop half the rise: -4 - 4 = -8°C; or recipe: 2/3 cup for 4 servings, make 6 (multiply by 6/4 = 1.5): (2/3) × (3/2) = 1 cup. To solve correctly, start at -6.5 + 4 = -2.5, rise was 4, drop by (3/2) × 4 = 6, so -2.5 - 6 = -8.5°C. A common error is sign error like -6.5 + 4 as -10.5 or treating drop as positive, wrong operation like adding instead of multiplying for the drop amount, or fraction wrong like 3/2 × 4 as 1.5 or 6 wrong, or arithmetic error like -2.5 - 6 as -8 or -9. To solve these, (1) identify operations: add rise, multiply rise by 3/2 then subtract that, (2) sequence correctly: temperature after rise, then drop calculated from rise, (3) apply sign rules for addition/subtraction with negatives, (4) convert fractions/decimals as needed, (5) track steps: -6.5+4=-2.5, 4×1.5=6, -2.5-6=-8.5, (6) interpret as final temperature below zero. Context clues like 'drops by 3/2 times the amount it rose' suggests multiply then subtract, mistakes in sign errors or misinterpreting the multiplier.

This question tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. Four operations with rationals: addition/subtraction (sign rules: same signs add magnitudes, different signs subtract, take larger's sign; subtraction as adding inverse: p-q=p+(-q)), multiplication/division (sign rules: same signs give positive, different signs give negative). Multi-step: track values through sequence (start $50, subtract $18.50→$31.50, add $30→$61.50, divide by 3→$20.50 each). Context: negative numbers for debt/loss/below zero/descent (meaningful negatives). For example, profit/loss average: Monday +$120, Tuesday -$45, Wednesday +$80, calculate (120+(-45)+80)÷3=(120-45+80)÷3=155÷3≈$51.67 average daily profit; or temperature: start -12°C, rise 8° (add): -12+8=-4°C, then drop half the rise: -4-4=-8°C; or recipe: 2/3 cup for 4 servings, make 6 (multiply by 6/4=1.5): (2/3)×(3/2)=1 cup. To solve, start with $35.00 - $6.75 = $28.25, then $28.25 + $12.50 = $40.75, then fee is (1/5)×$40.75 = $8.15, so $40.75 - $8.15 = $32.60. A common error is miscalculating the fee by multiplying 1/5 after a wrong balance, like forgetting to add the deposit and using $28.25 / 5 = $5.65, then $28.25 - $5.65 = $22.60 (not an option), or arithmetic error like 40.75 - 8.15 as 32.75 by misalignment of decimals. Solving: (1) identify all operations needed (read problem: add, subtract, multiply, divide?), (2) sequence operations (order matters: (a+b)÷c≠a+(b÷c)), (3) apply operation rules (sign rules for all four: same→positive quotient/product, different→negative), (4) convert forms if needed (2/3 and 1.5 to common form), (5) track through steps (running value after each operation), (6) interpret (result in context: -$5 means owes $5, -8°C means below zero). Operation priority: handle within parentheses or natural groups first, then multiply/divide, then add/subtract (or left-to-right if equal precedence). Context clues: "share" suggests divide, "total" suggests add/multiply, "difference" suggests subtract, "of" with fraction suggests multiply. Mistakes: operation order, sign errors (most common across all four operations), arithmetic with decimals/fractions, context interpretation (negative meaning).

This question tests solving real-world problems with all four operations on rational numbers (positive/negative integers/fractions/decimals), requiring multi-step calculations and context interpretation. Four operations with rationals: addition/subtraction (sign rules: same signs add magnitudes, different signs subtract, take larger's sign; subtraction as adding inverse: p-q=p+(-q)), multiplication/division (sign rules: same signs give positive, different signs give negative). Multi-step: track values through sequence (start $50, subtract $18.50→$31.50, add $30→$61.50, divide by 3→$20.50 each). Context: negative numbers for debt/loss/below zero/descent (meaningful negatives). For example, profit/loss average: Monday +$120, Tuesday -$45, Wednesday +$80, calculate (120+(-45)+80)÷3=(120-45+80)÷3=155÷3≈$51.67 average daily profit; or temperature: start -12°C, rise 8° (add): -12+8=-4°C, then drop half the rise: -4-4=-8°C; or recipe: 2/3 cup for 4 servings, make 6 (multiply by 6/4=1.5): (2/3)×(3/2)=1 cup. To solve, -0.6 + 0.35 = -0.25, then drain (2/5)×0.35 = 0.14, so -0.25 - 0.14 = -0.39 m. A common error is sign error like treating addition of 0.35 as subtraction to -0.6 - 0.35 = -0.95, then subtracting 0.14 to -1.09 (not an option), or wrong fraction like 2/5 × 0.35 as 0.7/5 = 0.14 but misapplying. Solving: (1) identify all operations needed (read problem: add, subtract, multiply, divide?), (2) sequence operations (order matters: (a+b)÷c≠a+(b÷c)), (3) apply operation rules (sign rules for all four: same→positive quotient/product, different→negative), (4) convert forms if needed (2/3 and 1.5 to common form), (5) track through steps (running value after each operation), (6) interpret (result in context: -$5 means owes $5, -8°C means below zero). Operation priority: handle within parentheses or natural groups first, then multiply/divide, then add/subtract (or left-to-right if equal precedence). Context clues: "share" suggests divide, "total" suggests add/multiply, "difference" suggests subtract, "of" with fraction suggests multiply. Mistakes: operation order, sign errors (most common across all four operations), arithmetic with decimals/fractions, context interpretation (negative meaning).