- A diver is at −621 meters relative to sea level (below the surface). The diver rises 4.75 meters, then descends 43 meter, then rises another 241 meters. What is the diver's final position relative to sea level?
Answer: −0.25 m
Starting at -6.5 meters, add the 4.75 meter rise to get -6.5 + 4.75 = -1.75 meters. Subtract the 0.75 meter descent to get -1.75 - 0.75 = -2.5 meters. Add the 2.25 meter rise to get -2.5 + 2.25 = -0.25 meters. Choice B comes from a sign error somewhere in the sequence of rises and descents. Choice C is close to the correct value but has the wrong sign, suggesting the diver ended above rather than below sea level. Choice D comes from missing the final rise of 2.25 meters. - A recipe serves 8 people and calls for 231 pounds of chicken. Lisa wants to make the recipe for 12 people. If chicken costs $4.80 per pound, how much will Lisa spend on chicken?
Answer: $16.80
The recipe needs to scale up by a factor of 12/8 = 1.5. Converting 2 1/3 pounds to an improper fraction gives 7/3 pounds, and multiplying by 1.5 (or 3/2) gives 7/3 x 3/2 = 7/2 = 3.5 pounds of chicken needed. At $4.80 per pound, the total cost is 3.5 x 4.80 = $16.80. Choice A uses the original recipe's chicken amount without scaling it up. Choice C comes from a miscalculation of the scaling factor. Choice D comes from an error in converting the mixed number 2 1/3 to a decimal or fraction.
- A smoothie recipe uses 2/3 cup of yogurt for 1 batch. A student makes 1.5 batches. How many cups of yogurt do they need?
Answer: 1 cup
To find the total yogurt needed, multiply the amount per batch by the number of batches: 2/3 cup x 1.5 = 2/3 x 3/2 = 1 cup, matching choice D. Choice A is just the per-batch amount, ignoring that 1.5 batches were made. Choice B comes from dividing instead of multiplying (2/3 divided by 3/2 = 4/9). Choice C simply restates the number of batches (1.5 = 1 1/2) instead of calculating the amount of yogurt.
- Estimate first, then solve exactly: A notebook is $3.60. A student buys 221 notebooks and uses a coupon for −$1.25 off the total cost (subtract $1.25). What is the final cost? (Round to the nearest cent.)
Answer: $7.75
This problem tests solving multi-step problems with rational numbers (whole, fraction, decimal, positive, negative), converting between forms strategically, and checking reasonableness. Multi-step with mixed forms: combine different formats ($3.60 decimal, 2 1/2 mixed=2.5, -$1.25 decimal), apply sequentially (multiply by 2.5, subtract 1.25). For this specific problem: 3.60 × 2.5 = 9.00, 9.00 - 1.25 = 7.75. The correct approach involves converting mixed number to decimal and applying operations in order. Common errors include operation order wrong (subtracting coupon before multiplying), conversion wrong (2 1/2 as 2.4), or skipping estimation. Strategy: (1) read carefully identifying all values and operations (list steps needed), (2) convert to consistent form if easier (all decimals), (3) execute step-by-step (track running total, don't skip), (4) estimate alongside (3.60≈4, ×2.5=10, -1=9 close but adjust to 7.75✓), (5) verify reasonable ($7.75 for 2.5 notebooks after discount makes sense), (6) check units (dollars stay dollars). Common mistakes: skipping estimation (missing unreasonable answers), form conversion errors, operation order (multiply before subtract).
- A door is 2721 inches wide. A towel bar that is 943 inches long is centered on the door. How far is each end of the towel bar from the nearest edge of the door? (Give your answer in inches.)
Answer: 887 in.
This problem tests solving multi-step problems with rational numbers (whole, fraction, decimal, positive, negative), converting between forms strategically, and checking reasonableness. Multi-step with mixed forms: combine different formats (27 1/2 mixed number, 9 3/4 mixed number), convert for operations (to decimals: 27.5, 9.75, or improper fractions), apply operations sequentially (subtract bar length, divide by 2). For this specific problem: door 27.5 in - bar 9.75 in = 17.75 in total space, divide by 2: 8.875 in = 8 7/8 in each side. The correct approach involves subtracting lengths then dividing equally, converting to consistent forms for accuracy. Common errors include operation order wrong (dividing before subtracting), conversion wrong (3/4 as 0.34), or unreasonable result not questioned (1 inch from edge obviously wrong for 9.75" bar on 27.5" door). Strategy: (1) read carefully identifying all values and operations (list steps needed), (2) convert to consistent form if easier (decimals or fractions—choose based on numbers), (3) execute step-by-step (track running total, don't skip), (4) estimate alongside (round: 28-10=18, 18/2=9 close to 8.875✓), (5) verify reasonable (about 9 inches each side makes sense), (6) check units (inches stay inches). Common mistakes: skipping estimation (missing unreasonable answers), form conversion errors (mixed numbers to decimals imprecise), operation order (subtract before divide). - A submarine starts at sea level and descends 45.8 meters. It then rises 1241 meters and descends again 8.75 meters. What is the submarine's final depth below sea level?
Answer: 42.3 meters below sea level
Convert 1241 to decimal: 12.25. Starting at 0, after descending 45.8 m: position is −45.8. After rising 12.25 m: −45.8+12.25=−33.55. After descending 8.75 m: −33.55−8.75=−42.3. The submarine is 42.3 meters below sea level. Choice A incorrectly subtracts the final descent. Choice B uses 12.5 instead of 12.25. Choice D makes an error in the middle calculation. - A recipe uses 32 cup of yogurt for 1 batch. Maya makes 1.5 batches, then accidentally spills 0.25 cup. How much yogurt does she have left from the amount she prepared? (Assume she prepared exactly what she needed for 1.5 batches before spilling.)
Answer: 0.75 cup
This problem tests solving multi-step problems with rational numbers, including whole numbers, fractions, decimals, positive and negative values, by converting between forms strategically and checking for reasonableness. For recipe scaling with 2/3 cup per batch, multiply by 1.5 (decimal or 3/2 fraction) to get (2/3) × (3/2) = 1 cup, then subtract spilled 0.25 cup to leave 0.75 cup. This involves multiplication first for batches, then subtraction, using fraction-decimal conversions. Correct steps ensure exact preparation for 1.5 batches before spilling. Common errors: wrong multiplication like 2/3 + 1.5, or subtracting before scaling. Strategy: convert to decimals or fractions, multiply then subtract, estimate 2/3≈0.67 ×1.5≈1, 1-0.25=0.75 exact, check reasonableness for partial batch remainder. Avoid order mistakes or not verifying with estimation. - Jake earns $18.50 per hour and works 621 hours on Saturday. He spends 52 of his earnings on gas and saves the rest. How much money does Jake save?
Answer: $72.15
Jake works 6.5 hours at $18.50 per hour, earning 18.50 x 6.5 = $120.25 total. He spends 2/5 of this on gas: 0.4 x 120.25 = $48.10. Subtracting the gas cost from his total earnings gives 120.25 minus 48.10, or $72.15 saved. Choice A shows only the amount spent on gas, not the amount saved. Choice D shows his total earnings before subtracting gas costs. Choice C comes from a miscalculation of the 2/5 gas amount.
- A student has $45.00 in a game account. They spend $18.50, then deposit $30.00. After that, they spend 41 of the new total. How much money is left in the account? (Round to the nearest cent.)
Answer: $42.38
This problem tests solving multi-step problems with rational numbers (whole, fraction, decimal, positive, negative), converting between forms strategically, and checking reasonableness. Multi-step with mixed forms: combine different formats (money $45 whole number, $18.50 decimal, $30 whole, 1/4 fraction), convert for operations (1/4=0.25 for easier decimal arithmetic, or convert all to fractions), apply operations sequentially (subtract 18.50, add 30, multiply by 1/4 for spending, subtract result—tracking values step-by-step). For this specific problem: start with $45, spend $18.50 (45-18.50=26.50), deposit $30 (26.50+30=56.50), spend 1/4 of 56.50 (56.50×0.25=14.125), subtract (56.50-14.125=42.375 rounded to $42.38). The correct approach involves performing the operations in sequence after conversions, ensuring the fraction is applied to the updated total before subtracting. Common errors include wrong operation order (adding before finding fraction 'of'), conversion mistakes (1/4 as 0.2 instead of 0.25), or arithmetic errors in intermediate steps propagating to an incorrect final amount. Strategy: (1) read carefully identifying all values and operations (list steps needed), (2) convert to consistent form if easier (all decimals here), (3) execute step-by-step (track running total, don't skip), (4) estimate alongside (round: 45-19=26, +30=56, 1/4 of 56=14, 56-14=42 close to 42.38✓), (5) verify reasonable ($42 remaining from $45 start with spending and deposit makes sense), (6) check units (dollars stay dollars). Common mistakes: skipping estimation (missing unreasonable answers), form conversion errors, operation order (fraction 'of' is multiply, must do before subtracting).
- A student runs 3.2 km on Monday. On Tuesday, they run 45 times Monday's distance. On Wednesday, they run 121 km less than Tuesday's distance. What total distance do they run over the three days?
Answer: 9.7 km
Tuesday's distance is 5/4 times Monday's 3.2 km, which is 3.2 x 1.25 = 4 km. Wednesday's distance is 1.5 km less than Tuesday's 4 km, which is 4 - 1.5 = 2.5 km. Adding all three days gives 3.2 + 4 + 2.5 = 9.7 km. Choice A comes from an error in one of the multiplication or subtraction steps. Choice B comes from adding instead of subtracting on Wednesday. Choice C comes from a smaller miscalculation in the Tuesday or Wednesday distance. - A water tank contains 43 of its capacity. After using 15.5 gallons, the tank is 21 full. What is the total capacity of the tank?
Answer: 62 gallons
This problem involves setting up an equation based on fractional parts of an unknown total capacity. When you see a word problem about fractions of a whole where the whole is unknown, think about defining a variable for the total and translating the given information into an equation.
Let's call the total capacity C gallons. Initially, the tank contains 43C gallons. After using 15.5 gallons, the tank has 43C−15.5 gallons remaining. We're told this remaining amount equals 21C gallons (half the total capacity).
Setting up the equation: 43C−15.5=21C
To solve, subtract 21C from both sides: 43C−21C=15.5
Converting to common denominators: 43C−42C=15.5, which gives us 41C=15.5
Therefore: C=15.5×4=62 gallons.
Choice A (60 gallons) might result from rounding 15.5 down to 15 before multiplying. Choice B (64 gallons) could come from incorrectly calculating 16×4 if you rounded 15.5 up. Choice C (58 gallons) might occur from computational errors in the fraction arithmetic or incorrectly setting up the initial equation.
Strategy tip: In fraction word problems involving an unknown total, always define your variable clearly and translate each piece of information into mathematical expressions before setting up your equation. Double-check by substituting your answer back into the original problem. - A hiker starts at an elevation of 450 m (above sea level). They descend 275 m, then climb 180 m. What is their final elevation?
Answer: 355 m
This problem tests solving multi-step problems with rational numbers (whole, fraction, decimal, positive, negative), converting between forms strategically, and checking reasonableness. Multi-step with mixed forms: all whole numbers here (450, -275 descend, +180 climb), apply operations sequentially (subtract 275, add 180). For this specific problem: 450 - 275 = 175, 175 + 180 = 355 m. The correct approach involves treating descent as subtraction and climb as addition in sequence. Common errors include sign error (adding descent as positive), or unreasonable result (negative elevation when starting high). Strategy: (1) read carefully identifying all values and operations (list steps needed), (2) convert to consistent form if easier (wholes here), (3) execute step-by-step (track running total, don't skip), (4) estimate alongside (450-300=150, +200=350 close to 355✓), (5) verify reasonable (ends positive above sea level—logical), (6) check units (meters stay meters). Common mistakes: skipping estimation (missing unreasonable answers), negative operations (descent is subtract), operation order wrong. - Estimate and then compute exactly to check reasonableness: A water tank is -1.8 liters (it needs 1.8 L to be full). First, 3 1/2 liters are added. Then 0.6 liter leaks out. Finally, 1/4 of the current amount of water is drained for cleaning. What is the final amount of water in the tank (in liters)?
Answer: 0.825 L
Start with the tank at negative 1.8 liters, meaning it needs 1.8 more liters to be full. Adding 3 and a half liters gives -1.8 + 3.5 = 1.7 liters. After the leak of 0.6 liters, the tank has 1.7 - 0.6 = 1.1 liters. Draining one fourth of that amount removes 0.25 x 1.1 = 0.275 liters, leaving 1.1 - 0.275 = 0.825 liters, which matches Choice B. Choice A has the correct value but the wrong sign, Choice C forgets the final draining step, and Choice D does not match any of the correct calculations.
- A bank account starts at −$12.50 (overdraft). A deposit of $35.00 is made, then a purchase of $18.40 is charged. After that, a fee equal to 51 of the remaining balance is deducted from the account. What is the final balance? Round to the nearest cent.
Answer: $3.28
Starting at -12.50, add the 35.00 deposit to get -12.50 + 35.00 = 22.50. Subtract the 18.40 purchase to get 22.50 - 18.40 = 4.10. The fee is 1/5 of this remaining balance, or 4.10 x 0.2 = 0.82, and subtracting the fee gives 4.10 - 0.82 = 3.28. Choice B and choice C both come from sign or ordering errors somewhere in the multi-step process. Choice D comes from applying the fee at the wrong point in the sequence of transactions.
- At sunrise, the temperature is −8∘C. By lunchtime it rises 15∘C, and in the afternoon it drops 3.5∘C. What is the final temperature?
Answer: 3.5∘C
This problem tests solving multi-step problems with rational numbers, including whole numbers, fractions, decimals, positive and negative values, by converting between forms strategically and checking for reasonableness. Multi-step with mixed forms involve combining temperatures like -8 whole negative, +15 whole positive, -3.5 decimal, requiring careful sign handling in addition and subtraction to track the net change. For this temperature scenario, start at -8°C, add 15°C to get 7°C, then subtract 3.5°C to reach 3.5°C, ensuring signs are applied correctly for rises and drops. The correct multi-step process uses addition for rises and subtraction for drops, with decimal arithmetic to find the positive final temperature. Errors might include sign mishandling, like treating the drop as addition of negative incorrectly to get -3.5°C, or ignoring the initial negative. Strategy: identify operations (add for rise, subtract for drop), use a number line for negatives, compute step-by-step, estimate -8+15≈7, 7-4≈3 close to 3.5, and check reasonableness like starting cold and net warming to positive. Common mistakes: forgetting negative rules or not estimating to catch unreasonable sub-zero finals after significant rise. - A student has $20.00 on a lunch card. They buy lunch for $6.75, then they add $12.50. After that, a school fee of 52 of the current balance is taken out. How much money is left? Round to the nearest cent.
Answer: $15.45
Start with $20.00, subtract the $6.75 lunch purchase to get $13.25, then add $12.50 to get $25.75. The school fee is 2/5 of this balance, which is 2/5 x $25.75 = $10.30. Subtracting the fee leaves $25.75 minus $10.30, or $15.45. Choice A shows the balance before the fee is subtracted. Choice B shows only the fee amount, not the remaining balance. Choice D comes from a small rounding or subtraction slip near the final step.
- A diver is at −12 meters relative to sea level. They rise 7.5 meters, then drop 49 meters. What is their final position relative to sea level?
Answer: −6.75 m
This problem tests solving multi-step problems with rational numbers (whole, fraction, decimal, positive, negative), converting between forms strategically, and checking reasonableness. Multi-step with mixed forms: combine different formats (-12 whole negative, +7.5 decimal, -9/4 fraction= -2.25), apply sequentially (add 7.5, subtract 2.25). For this specific problem: -12 + 7.5 = -4.5, -4.5 - 2.25 = -6.75 m. The correct approach involves converting fraction to decimal and handling negative positions properly. Common errors include sign error (-12 +7.5 as -19.5), conversion wrong (9/4 as 2.5), or unreasonable result (positive when diving deeper). Strategy: (1) read carefully identifying all values and operations (list steps needed), (2) convert to consistent form if easier (all decimals), (3) execute step-by-step (track running total, don't skip), (4) estimate alongside (-12+8=-4, -4-2=-6 close to -6.75✓), (5) verify reasonable (ends below sea level—logical for diver), (6) check units (meters stay meters). Common mistakes: skipping estimation (missing unreasonable answers), form conversion errors, negative operations (rise is add, drop is subtract). - A trail mix recipe calls for 141 cups of nuts, 0.75 cups of dried fruit, and 83 cups of chocolate chips. If Maya wants to make 2.5 times the recipe but only has 2 cups of nuts available, how much more nuts does she need?
Answer: 1.125 cups
This problem combines mixed numbers, decimals, and fractions while testing your ability to scale recipes and calculate differences. When you see questions mixing different number formats, convert everything to the same form first to avoid confusion.
To find how much more nuts Maya needs, you must first determine how many nuts the scaled recipe requires. The original recipe calls for 141 cups of nuts. Converting to a decimal: 141=1.25 cups. When Maya makes 2.5 times the recipe, she needs 1.25×2.5=3.125 cups of nuts total. Since she has 2 cups available, she needs 3.125−2=1.125 more cups.
Choice A (3.125 cups) represents the total amount of nuts needed for the scaled recipe, not the additional amount required. This is a common trap where students stop calculating before finding the final answer. Choice B (0.875 cups) likely comes from incorrectly calculating 2.5−1.25=1.25, then subtracting 2−1.25=−0.75, and taking the absolute value incorrectly. Choice C (1.25 cups) is simply the original amount of nuts in the recipe, showing the student forgot to scale up by 2.5.
The correct answer is D (1.125 cups).
Strategy tip: In multi-step word problems, write out each calculation step clearly: (1) find the scaled requirement, (2) subtract what's available. Also, when mixing fractions and decimals, convert everything to decimals first to avoid computational errors. - A door is 2721 inches wide. A towel bar that is 943 inches long is centered on the door. How far is each end of the bar from the nearest edge of the door?
Answer: 887 inches
This problem tests solving multi-step problems with rational numbers, including whole numbers, fractions, decimals, positive and negative values, by converting between forms strategically and checking for reasonableness. For centering with mixed numbers like door 27 1/2 (fraction) and bar 9 3/4 (fraction), convert to improper fractions or decimals, subtract lengths, then divide by 2 for each side. Here, 27.5 - 9.75 = 17.75, then 17.75 / 2 = 8.875 inches, or in fractions 55/2 - 39/4 = (110/4 - 39/4) = 71/4, then (71/4)/2 = 71/8 = 8 7/8. Correct steps ensure subtraction first, then equal division, converting back to mixed numbers. Errors include dividing before subtracting or conversion slips like 1/2 as 0.6. Strategy: convert to decimals for ease, subtract, divide, estimate 28-10=18, 18/2=9 close to 8.875, check reasonableness for balanced spacing. Common mistakes: operation order wrong or not estimating to spot tiny spacings. - A student has $45.00 in a wallet. They spend $18.50, then deposit $30.00. Next, they spend 41 of the money they have at that point. How much money is left? (Round to the nearest cent.)
Answer: $42.38
This problem tests solving multi-step problems with rational numbers, including whole numbers, fractions, decimals, positive and negative values, by converting between forms strategically and checking for reasonableness. In multi-step scenarios with mixed forms, such as starting with $45 whole number, subtracting $18.50 decimal, adding $30 whole, then spending 1/4 fraction of the remainder, convert fractions to decimals for easier arithmetic, like 1/4=0.25, and apply operations sequentially while tracking values step-by-step. For this specific problem, start with $45, subtract $18.50 to get $26.50, add $30 to reach $56.50, then spend $56.50 × 0.25 = $14.125 (rounds to $14.13), and subtract to find $56.50 - $14.125 = $42.375, which rounds to $42.38. The correct approach involves proper order of operations and accurate conversions, ensuring the final amount is calculated after all steps. Common errors include incorrect operation order, like multiplying before adding the deposit, or conversion mistakes such as treating 1/4 as 0.20. Strategy: read carefully to list steps, convert to decimals for consistency, execute step-by-step with running totals, estimate like 45-19+30=56, 56/4=14 spent, 56-14=42 close to $42.38, and verify reasonableness with net spending and deposit. Avoid mistakes like skipping estimation or arithmetic errors in subtraction.
