- A water tank contains 840 gallons. Water is being drained at 12 gallons per minute while simultaneously being filled at 7 gallons per minute. How long will it take to empty the tank completely?
Answer: 168 minutes
When you encounter a problem involving simultaneous rates working in opposite directions, you need to find the net rate of change. This tank has water flowing both out and in at the same time, so you must determine the overall effect.
The tank is draining at 12 gallons per minute while being filled at 7 gallons per minute. Since these work in opposite directions, subtract to find the net drainage rate: 12−7=5 gallons per minute. The tank is losing water at a net rate of 5 gallons per minute.
To find how long it takes to empty 840 gallons at this net rate, divide the total volume by the net rate: 5840=168 minutes. This confirms that answer B is correct.
Looking at the incorrect choices: A (140 minutes) results from incorrectly dividing 840 by 6, perhaps from miscalculating the net rate as 12−7+1=6. C (175 minutes) comes from dividing 840 by 4.8, which suggests an error in calculating the net rate. D (210 minutes) results from dividing 840 by 4, which would happen if you mistakenly calculated the net rate as 12−7−1=4 or made another arithmetic error.
Remember: in simultaneous rate problems, always identify which rates work together and which work against each other. Subtract opposing rates to find the net effect, then use that net rate in your time calculations. Double-check your arithmetic on the rate calculation since small errors there multiply into large errors in the final answer. - A farmer plants corn in a field that measures 120 feet by 80 feet. If each corn plant needs 4 square feet of space and seeds cost $0.25 each, what is the total cost for seeds?
Answer: $600
This is a multi-step area and cost calculation problem that tests your ability to work through sequential operations methodically.
Start by finding the total area of the field: 120×80=9,600 square feet. Next, determine how many corn plants can fit by dividing the total area by the space each plant needs: 9,600÷4=2,400 plants. Finally, calculate the total seed cost: 2,400×$0.25=$600. This confirms answer choice B.
Let's examine why the other options are incorrect. Choice A ($480) likely comes from a calculation error, possibly multiplying 2,400 plants by $0.20 instead of $0.25, or making an arithmetic mistake when computing the area. Choice C ($720) might result from using the wrong area calculation or incorrectly applying the space requirement—perhaps dividing by 3 instead of 4, giving 3,200 plants, then multiplying by $0.225. Choice D ($2,400) represents a common trap: this is actually the number of plants needed, but the question asks for the total cost, not the quantity of seeds.
When tackling multi-step word problems like this, always identify what the question is actually asking for in the final step. Write down each calculation clearly: area first, then number of plants, then total cost. This prevents you from stopping early or confusing intermediate results (like the number of plants) with the final answer (total cost). - A school cafeteria serves 450 students lunch. Each student receives 2 slices of pizza, and each pizza is cut into 8 slices. If pizzas cost $12 each and the cafeteria gets a 15% bulk discount, what is the total cost for pizza?
Answer: $1,152.60
Work in order: slices, whole pizzas, cost, discount.
The students need 450×2=900 slices. At 8 slices per pizza that is 8900=112.5 pizzas, and since the cafeteria cannot buy half a pizza, it must buy 113. Before the discount, 113 pizzas cost 113×12=1,356 dollars. A 15% discount means paying 85%: 1,356×0.85=1,152.60, so the total is $1,152.60.
The amount $1,147.50 is 112.5×12×0.85, which pays for half a pizza that cannot be bought. The amount $1,350.00 is 112.5×12 with no discount at all. The amount $1,224.00 is the discounted price of 120 pizzas, more than are needed.
Tip: when items come whole, round the count up before you multiply by the price. - A factory produces 1,440 widgets per day. Each widget requires 3 bolts, and bolts are sold in packages of 24. If the factory operates 5 days per week, how many packages of bolts are needed for one week of production?
Answer: 900 packages
This is a multi-step word problem that tests your ability to work through a chain of calculations systematically. When you encounter problems involving production rates and packaging, break down each piece of information and work step by step.
First, calculate the total widgets needed for one week: 1,440 widgets/day×5 days=7,200 widgets
Next, find the total bolts required: 7,200 widgets×3 bolts/widget=21,600 bolts
Finally, determine how many packages are needed: 24 bolts/package21,600 bolts=900 packages
This confirms that A) 900 packages is correct.
Looking at the wrong answers: B) 975 packages likely results from miscalculating the daily widget production or making an arithmetic error in the division. C) 1,080 packages might come from forgetting to multiply by the 3 bolts per widget, instead dividing 7,200 widgets directly by 24. D) 1,200 packages could result from incorrectly using 6 days instead of 5, or making other computational errors along the way.
Strategy tip: In multi-step word problems, write down each intermediate result clearly. Many students make errors by trying to do too many calculations mentally or by rushing through the steps. Also, double-check that you've used all the given information—if a number appears in the problem, it's usually essential to the solution. - A rectangular garden has a length of 24 feet and a width of 18 feet. If fencing costs $12 per foot and there is a 6% sales tax, what is the total cost to fence the entire perimeter?
Answer: $1,068.48
This problem combines geometry (perimeter) with multi-step calculations involving taxes. When you see questions about fencing or bordering a rectangular area, you're calculating perimeter, then applying cost factors.
First, find the perimeter of the rectangular garden. The perimeter formula is P=2l+2w, where l is length and w is width. With a length of 24 feet and width of 18 feet: P=2(24)+2(18)=48+36=84 feet.
Next, calculate the pre-tax cost. At $12 per foot for 84 feet: 84×12=$1,008.
Finally, add the 6% sales tax. The tax amount is 1,008×0.06=$60.48. The total cost is 1,008+60.48=$1,068.48.
Looking at the wrong answers: Choice A ($954.24) appears to use an incorrect perimeter calculation, possibly using area instead of perimeter. Choice B ($1,008.00) gives you the pre-tax cost but forgets to include the sales tax entirely. Choice D ($1,152.00) suggests a calculation error, possibly using the wrong tax rate or making an arithmetic mistake.
The correct answer is C.
Study tip: Multi-step word problems like this are common on the ISEE. Always identify what you're solving for first (here: perimeter), then apply additional factors (cost per unit, taxes) in sequence. Don't rush—each step builds on the previous one, so one error cascades through your entire solution. - A bakery produces cupcakes using a recipe that makes 36 cupcakes from 4.5 cups of flour. If they want to make 288 cupcakes and flour costs $1.20 per cup, what will the flour cost?
Answer: $43.20
This problem tests your ability to handle multi-step proportional reasoning with unit rates. When you see a recipe or production problem asking for scaling and costs, break it down into clear steps: find the rate, scale up, then calculate the final cost.
Start by finding how much flour is needed per cupcake: 36 cupcakes4.5 cups=0.125 cups per cupcake. For 288 cupcakes, you'll need 288×0.125=36 cups of flour. At $1.20 per cup, the total cost is 36×1.20=$43.20.
Answer A ($28.80) represents a common error where students incorrectly calculate the scaling factor. They might think 288 ÷ 36 = 8, then multiply 4.5 × 8 = 36, but then mistakenly use $0.80 per cup instead of $1.20. Answer B ($32.40) occurs when students correctly find 36 cups needed but multiply by $0.90 per cup, possibly misreading the given cost. Answer C ($36.00) is the trap for students who correctly calculate 36 cups of flour but forget to multiply by the cost per cup—they're giving the quantity, not the price.
The correct answer is D ($43.20).
Remember this pattern: in multi-step word problems, write down each intermediate result clearly. First find the unit rate, then scale to the new quantity, then apply any cost calculations. Don't skip steps mentally—the test writers design wrong answers to catch mental math errors at each stage. - A company manufactures 1,800 items per day using 15 workers. If they hire 5 more workers and each worker's productivity increases by 10%, how many items will they produce per day?
Answer: 2,640 items
When you encounter productivity problems involving changes in both workforce size and individual output, you need to track both variables systematically to avoid calculation errors.
Start by finding the current productivity per worker: 15 workers1,800 items=120 items per worker per day
Next, calculate the new conditions. With 5 additional workers, the company now has 20 workers total. Each worker's productivity increases by 10%, so each worker now produces: 120×1.10=132 items per day
Therefore, total daily production becomes: 20 workers×132 items per worker=2,640 items
This confirms answer A is correct.
Looking at the wrong answers: B (2,772 items) likely results from miscalculating the 10% productivity increase or making an arithmetic error in the final multiplication. C (2,880 items) appears to come from correctly calculating 20 workers × 144 items, but 144 would represent a 20% increase (not 10%) in individual productivity. D (3,168 items) suggests applying the 10% increase to the final total production rather than to individual worker productivity—a common conceptual error.
Strategy tip: In multi-step productivity problems, always identify what's changing (workforce size, individual output, or both) and handle each change separately before combining them. Double-check that percentage increases are applied to the right base value—individual worker productivity, not total company output. - A swimming pool is being filled with water. In the first hour, 120 gallons are added. In each subsequent hour, 25% more water is added than in the previous hour. How many gallons are added in the fourth hour?
Answer: 234.4 gallons
This question tests exponential growth patterns, where each term increases by a fixed percentage. When you see "25% more than the previous hour," you're dealing with a growth factor of 1.25 (since 100% + 25% = 125% = 1.25).
Let's track the water added each hour systematically. Hour 1 starts with 120 gallons. For each subsequent hour, multiply the previous hour by 1.25:
- Hour 1: 120 gallons
- Hour 2: 120×1.25=150 gallons
- Hour 3: 150×1.25=187.5 gallons
- Hour 4: 187.5×1.25=234.375 gallons
Rounding to one decimal place gives us 234.4 gallons for the fourth hour.
Looking at the wrong answers: (A) 187.5 gallons is actually the amount added in the third hour, not the fourth—this catches students who miscounted or stopped one step early. (B) 210.9 gallons doesn't follow any logical pattern from this sequence and likely results from calculation errors in the multiplication. (D) 292.5 gallons is too large and suggests a student may have applied the 25% increase incorrectly, perhaps adding 25% to the wrong base value or making an error in the exponential pattern.
The correct answer is (C) 234.4 gallons.
Strategy tip: For percentage growth problems, always write out each step clearly rather than trying to jump ahead. Convert percentages to decimal multipliers immediately (25% increase = ×1.25) to avoid arithmetic mistakes. - A store sells notebooks in packs of 5 for $8 and pens in packs of 12 for $15. If a school needs 85 notebooks and 156 pens, what is the minimum total cost?
Answer: $331
When you encounter word problems involving bulk purchases, you need to think about buying in fixed package sizes rather than individual items. Since you can't buy partial packages, you'll often need to purchase more than the exact amount required.
For notebooks: You need 85 notebooks, and they're sold in packs of 5 for $8. Divide 85 by 5 to get exactly 17 packs needed. Cost: 17×8=$136.
For pens: You need 156 pens, and they're sold in packs of 12 for $15. Divide 156 by 12 to get exactly 13 packs needed. Cost: 13×15=$195.
Total minimum cost: 136+195=$331, which is answer A.
The incorrect answers likely represent common calculation errors. Answer B ($356) might result from miscalculating one of the multiplication steps or adding an extra pack unnecessarily. Answer C ($376) could come from errors in division when determining how many packs are needed, perhaps rounding up when exact division wasn't required. Answer D ($391) represents a more significant computational error, possibly from misreading the pack sizes or prices.
The key insight here is that both 85 and 156 divide evenly into their respective pack sizes, so no rounding up is necessary. Always check your division carefully in these problems—sometimes you'll need to round up to the next whole pack, but sometimes (like here) the division works out perfectly. Double-check your arithmetic, especially when multiplying the number of packs by their individual costs. - A gym membership costs $45 per month with a one-time enrollment fee of $75. If Sarah pays for 8 months and receives a 10% discount on the monthly fees only, what is her total cost?
Answer: $399
When you encounter multi-step cost problems with discounts, break down each component separately and pay close attention to what the discount applies to.
Let's calculate Sarah's total cost step by step. She has two types of charges: monthly fees and a one-time enrollment fee. The monthly cost is $45 × 8 months = $360. Since she receives a 10% discount on monthly fees only, her discounted monthly total is $360 × 0.90 = $324. The enrollment fee of $75 has no discount applied. Therefore, her total cost is $324 + $75 = $399.
Looking at the wrong answers: Choice A ($384) likely comes from applying the 10% discount to the entire amount ($435 × 0.90 = $391.50, though this doesn't match exactly, suggesting a calculation error). Choice C ($408) appears to result from calculating the discount incorrectly, perhaps subtracting only $27 instead of $36 from the monthly fees ($360 - $27 = $333, plus $75 = $408). Choice D ($435) is what you'd get if you ignored the discount entirely ($360 + $75 = $435).
The key strategy here is to read carefully what the discount applies to. Test makers often create traps by having students either apply discounts too broadly (to everything) or miscalculate the discount amount. Always identify each cost component first, apply discounts only where specified, then sum everything up. This systematic approach prevents the common error of rushing through multi-step calculations.
- A bookstore orders 144 books at $15 each. They receive a 20% trade discount but must pay 8% sales tax on the discounted price. What is the total amount paid?
Answer: $1,866.24
When you encounter multi-step discount and tax problems, work through each calculation sequentially, applying percentages to the correct base amounts at each stage.
Start with the original purchase: 144 books × $15 = $2,160. Next, apply the 20% trade discount. The discount amount is $2,160 × 0.20 = $432, so the discounted price becomes $2,160 - $432 = $1,728.
Now here's the crucial step: the 8% sales tax applies to the already-discounted price of $1,728, not the original $2,160. Calculate the tax: $1,728 × 0.08 = $138.24. The final total is $1,728 + $138.24 = $1,866.24.
Choice A ($1,866.24) correctly follows this sequence of discount first, then tax on the discounted amount. Choice B ($1,944.00) likely represents applying only the 8% tax to the discounted price without adding it back, or miscalculating the discount. Choice C ($2,099.52) appears to calculate the 8% sales tax on the original $2,160 price before applying the discount, which reverses the proper order. Choice D ($2,332.80) seems to add both the 20% and 8% as increases to the original price, completely misunderstanding that one is a discount and treating both as additional charges.
Remember: when problems involve both discounts and taxes, discounts typically apply first to reduce the base price, then taxes apply to that reduced amount. Always read carefully to confirm the order of operations and which amount serves as the base for each percentage calculation.
- A pizza parlor uses 3 ounces of cheese per pizza. If cheese costs $4.80 per pound and they make 64 pizzas in one day, what is the daily cost for cheese?
Answer: $57.60
This is a multi-step unit conversion and calculation problem that tests your ability to work with different units of measurement while solving a real-world scenario.
To find the daily cheese cost, you need to calculate the total ounces of cheese used, convert to pounds, then multiply by the cost per pound. First, find total cheese usage: 64 pizzas × 3 ounces per pizza = 192 ounces. Next, convert ounces to pounds: 192 ounces ÷ 16 ounces per pound = 12 pounds. Finally, calculate the cost: 12 pounds × $4.80 per pound = $57.60.
Looking at the wrong answers: Choice B ($61.44) likely comes from using an incorrect conversion factor, perhaps dividing by 15 instead of 16 when converting ounces to pounds. Choice C ($64.80) might result from forgetting the unit conversion entirely and treating the 192 ounces as if they were pounds (192 ÷ 16 = 12, but then multiplying 192 × $4.80 ÷ 15). Choice D ($76.80) could come from multiplying 192 by $4.80 and then dividing by 12, mixing up the conversion steps.
The correct answer is A ($57.60).
Strategy tip: On unit conversion problems, always identify what units you're starting with and what units you need to end up with. Write out the conversion factor (16 ounces = 1 pound) and double-check that your units cancel properly in your calculations. This systematic approach prevents the unit mix-ups that create most wrong answer choices.
- A rectangular parking lot measures 80 meters by 60 meters. If each parking space is 2.5 meters wide and 5 meters long, and 15% of the lot must remain as driving lanes, how many parking spaces can fit?
Answer: 326 spaces
When you encounter area and space allocation problems, you need to work systematically through multiple steps: find usable area, determine space per unit, then calculate how many units fit.
Start by finding the total lot area: 80×60=4800 square meters. Since 15% must remain as driving lanes, only 85% is available for parking: 4800×0.85=4080 square meters of usable space.
Next, calculate the area of each parking space: 2.5×5=12.5 square meters per space.
Finally, divide the usable area by the space per car: 4080÷12.5=326.4 spaces. Since you can't have partial spaces, this gives you 326 complete parking spaces.
Looking at the wrong answers: Choice B (345 spaces) likely comes from a calculation error or using the wrong percentage for driving lanes. Choice C (408 spaces) suggests someone calculated 4080÷10 instead of 4080÷12.5, possibly confusing the dimensions. Choice D (480 spaces) represents 4800÷10, meaning the student forgot to account for driving lanes entirely and used incorrect space dimensions.
The key strategy for these multi-step area problems is to work methodically: total area → usable area → individual unit area → final division. Always double-check that you've applied all given constraints (like the 15% driving lane requirement) and used the correct dimensions throughout your calculations. - Marcus saves $25 each week for 8 weeks. He then spends $75 on a video game and divides the remaining money equally among 5 charity donations. How much does he donate to each charity?
Answer: $25 per charity
This is a multi-step word problem that tests your ability to work through operations in the correct sequence. When you encounter problems with multiple steps, always identify what you're solving for first, then work backwards to see what information you need.
Marcus saves 25×8=$200 over 8 weeks. After spending $75 on a video game, he has $200−$75=$125 remaining. Since he divides this equally among 5 charities, each charity receives $125÷5=$25. This confirms answer choice A is correct.
Let's examine why the other answers are wrong. Answer choice B ($29 per charity) would mean Marcus donated $29×5=$145 total, but he only had $125 left after buying the game. Answer choice C ($35 per charity) would require $35×5=$175 in donations, which exceeds his remaining money by $50. Answer choice D ($40 per charity) would need $40×5=$200 for donations alone, ignoring the $75 he spent on the video game entirely.
The key strategy here is to work systematically through each step: calculate total savings first, subtract expenses second, then divide what remains. Many students make errors by rushing or mixing up the order of operations. Always double-check that your final answer makes sense in the context—if Marcus only had $125 left, his per-charity donation must be reasonable given that amount divided by 5. - A delivery truck travels 285 miles one way and uses 15 gallons of fuel on that one-way trip. If fuel costs $3.40 per gallon and the truck makes 4 round trips, what is the total fuel cost for all trips?
Answer: $408.00
Each one-way trip uses 15 gallons, so start from the number of one-way trips.
Four round trips are 4×2=8 one-way trips. The fuel used is 8×15=120 gallons, and the cost is 120×3.40=408 dollars, or $408.00. (The mileage, 19 miles per gallon, gives the same result: 8×285=2,280 miles and 2,280÷19=120 gallons.)
The amount $204.00 counts each round trip as one 15-gallon trip, using only 60 gallons. The amount $306.00 is 90 gallons, the fuel for 3 round trips. The amount $612.00 is 180 gallons, as if there were 6 round trips.
Tip: a round trip is two one-way trips; double before multiplying by the fuel per trip. - A rectangular swimming pool is 25 meters long and 15 meters wide. If the pool is filled to a depth of 1.5 meters and water costs $0.003 per liter, what is the cost to fill the pool?
Answer: $1,687.50
This problem combines volume calculations with unit conversions and cost analysis. When you see a word problem involving a three-dimensional container and pricing, work systematically through: volume → unit conversion → cost calculation.
First, find the pool's volume using the formula for a rectangular prism: length × width × height. The pool measures 25 meters × 15 meters × 1.5 meters = 562.5 cubic meters.
Next, convert to liters since the cost is given per liter. Remember that 1 cubic meter = 1,000 liters, so 562.5 cubic meters × 1,000 = 562,500 liters.
Finally, calculate the cost: 562,500 liters × $0.003 per liter = $1,687.50.
Looking at the wrong answers: Choice B ($1,875.00) likely comes from a calculation error, possibly using incorrect dimensions or forgetting part of the conversion. Choice C ($2,025.00) and Choice D ($2,250.00) represent larger errors, possibly from incorrectly converting units (maybe confusing cubic meters with liters directly) or using wrong cost calculations.
The correct answer is A) $1,687.50.
Strategy tip: For multi-step word problems like this, write down each step clearly: identify what you're solving for, convert all units to match the given rate, then multiply. Double-check your unit conversions—this is where most errors occur. Always verify that your final answer makes intuitive sense given the pool size and water cost.
- A bakery produces 144 muffins in the morning and 216 muffins in the afternoon. If the muffins are packed into boxes of 18, and 3 boxes are set aside for the staff, how many boxes are available for sale?
Answer: 17 boxes
This is a multi-step word problem that tests your ability to break down a real-world scenario into mathematical operations. When you see problems involving production, packaging, and distribution, work through each step systematically.
First, find the total number of muffins produced: 144+216=360 muffins. Next, determine how many boxes this creates when packed 18 per box: 360÷18=20 boxes total. Finally, subtract the boxes set aside for staff: 20−3=17 boxes available for sale.
Looking at the wrong answers: Choice B (20 boxes) represents the total number of boxes before removing the staff allocation—this is the trap for students who forget the final step. Choice C (23 boxes) incorrectly adds the staff boxes instead of subtracting them (20+3=23), which doesn't make logical sense since boxes set aside reduce availability. Choice D (26 boxes) likely comes from calculation errors, possibly adding the staff boxes to the total muffin count before dividing, or other arithmetic mistakes in the multi-step process.
The correct answer is A (17 boxes).
For multi-step word problems like this, always identify what the question is ultimately asking, then work backward to determine what information you need. Write down each step clearly: total production → total boxes → boxes after staff allocation. This prevents you from stopping too early or making logical errors about whether quantities should be added or subtracted.
