Estimating Whole Number Results - Accuplacer Arithmetic

1. An employee calculates the total cost of 42 items, each priced at $198, as $6,316. Without performing the exact calculation, which statement best describes the employee's result?
   Answer: The result is unreasonable because an estimate of $40 \times 200$ is $8,000. To check for reasonableness, round the numbers and estimate the product. Round 42 down to 40 and 198 up to 200. The estimated product is $40 \times 200 = 8,000$. The employee's result of $6,316 is significantly lower than this reasonable estimate, so it is unreasonable. While the last digit is correct ($2 \times 8 = 16$), this alone does not make the entire result reasonable.
2. A manufacturing plant calculates monthly labor costs. Shift 1 employs 247 workers at $23.50 per hour for 168 hours, Shift 2 employs 189 workers at $25.75 per hour for 168 hours, and Shift 3 employs 156 workers at $27.25 per hour for 168 hours. What is the best estimate of total monthly labor costs?
   Answer: $2,600,000 Estimate each shift for 168 ≈ 170 hours: Shift 1: 247 ≈ 250 workers × $23.50 ≈ $24 × 170 = $1,020,000; Shift 2: 189 ≈ 190 workers × $25.75 ≈ $26 × 170 = $838,200; Shift 3: 156 ≈ 160 workers × $27.25 ≈ $27 × 170 = $734,400. Total: $1,020,000 + $838,200 + $734,400 = $2,592,600 ≈ $2,600,000. Choice A underestimates hours or wages significantly. Choice B underestimates worker counts. Choice D overestimates wages to $30+ per hour.
3. A retail store manager estimates quarterly inventory costs. Department 1 orders $12,847 worth of merchandise, Department 2 orders $8,456 worth, Department 3 orders $15,234 worth, and Department 4 orders $6,789 worth. If they restock 3 times per quarter, what is the approximate total quarterly cost?
   Answer: $130,000 Sum initial orders: $12,847 + $8,456 + $15,234 + $6,789 = $43,326 ≈ $43,000. Quarterly cost (3 restocks): $43,000 × 3 = $129,000 ≈ $130,000. Choice A underestimates the sum (closer to $40,000). Choice C overestimates the sum (closer to $47,000). Choice D significantly overestimates, suggesting $50,000 per restock.
4. A delivery service calculates daily package volume. Monday: 1,234 packages, Tuesday: 2,156 packages, Wednesday: 1,887 packages, Thursday: 2,443 packages, Friday: 1,956 packages. If they charge $8.75 per package, approximately how much revenue did they generate this week?
   Answer: $86,000 Total packages: 1,234 + 2,156 + 1,887 + 2,443 + 1,956 = 9,676 ≈ 9,700. Revenue: 9,700 × $8.75 ≈ 9,700 × $9.00 = $87,300 ≈ $86,000. Choice A underestimates by using $8.50 price or 9,500 packages. Choice C overestimates packages to 10,000. Choice D overestimates both packages and price significantly.
5. A runner aims to complete a 42,195-meter race. She runs at an average pace of 198 meters per minute. What is the most reasonable estimate for the total time, in minutes, it will take her to finish the race?
   Answer: 210 To estimate the time, divide the distance by the pace. Round the distance 42,195 meters down to 42,000 meters. Round the pace 198 meters per minute up to 200 meters per minute. The estimated time is $42,000 \div 200$. This simplifies to $420 \div 2 = 210$ minutes.
6. A painter needs to cover a wall that is 19 feet tall and 88 feet long. A can of paint covers 410 square feet. What is the best estimate for the number of paint cans needed to cover the wall?
   Answer: 4 First, estimate the area of the wall. Round 19 feet to 20 feet and 88 feet to 90 feet. The estimated area is $20 \times 90 = 1,800$ square feet. Next, estimate the coverage per can by rounding 410 square feet to 400. Finally, divide the estimated area by the estimated coverage per can: $1,800 \div 400 = 4.5$. Since this is an estimation problem asking for the best estimate, 4 is the closest whole number to 4.5.
7. A pipeline can transport 195 gallons of oil per minute. What is the best estimate for the number of gallons it can transport in a full 24-hour day? (Note: There are 60 minutes in an hour).
   Answer: 280,000 This is a multi-step problem. First, estimate the gallons per hour. Round 195 gallons/minute to 200. Multiply by 60 minutes/hour: $200 \times 60 = 12,000$ gallons per hour. Second, estimate the gallons per day. Multiply the hourly rate by 24 hours. $12,000 \times 24$. You can calculate this as $12 \times 24 \times 1000 = 288 \times 1000 = 288,000$. The closest answer choice is 280,000. The actual value is $195 \times 60 \times 24 = 280,800$.
8. A large water tank holds 35,870 gallons. If the water is drained using a pump that removes 58 gallons per minute, which calculation provides the closest estimate for the number of minutes it will take to drain the tank?
   Answer: $36,000 \div 60$ To estimate the time, round the total gallons and the rate to compatible numbers. Round 35,870 gallons up to 36,000. Round 58 gallons per minute up to 60. The division $36,000 \div 60$ is easy to compute ($3600 \div 6 = 600$) and provides a close estimate. The actual answer is approximately 618 minutes, making 600 a very reasonable estimation.
9. A city has a budget of $12,125,000. It plans to spend $7,985,000 on infrastructure and $2,015,000 on public safety. Approximately how much of the budget remains?
   Answer: $2,000,000 First, estimate the total spending. Round $7,985,000 to $8,000,000 and $2,015,000 to $2,000,000. The estimated total spending is $8,000,000 + $2,000,000 = $10,000,000. Next, round the total budget to $12,000,000. Subtract the estimated spending from the estimated budget: $12,000,000 - $10,000,000 = $2,000,000.
10. A restaurant chain is calculating weekly food costs across all locations. Location A spends $3,847 per week, Location B spends $5,234 per week, Location C spends $2,156 per week, and Location D spends $4,891 per week. To estimate their monthly food costs (4 weeks), which calculation is most reasonable?
    Answer: $16,000 × 4 = $64,000 When you encounter estimation problems involving multiple values, start by finding the total of all given amounts, then round strategically to make calculations manageable. First, let's add up the weekly costs for all locations: $3,847 + $5,234 + $2,156 + $4,891 = $16,128. To estimate monthly costs, you multiply this weekly total by 4 weeks: $16,128 × 4 = $64,512. Now compare this exact answer to the given options. Choice A estimates the weekly total as $16,000, giving $16,000 × 4 = $64,000. This is remarkably close to our precise calculation of $64,512 – only off by about $500, making it the most reasonable estimate. Choice B ($15,000 × 4 = $60,000) underestimates the weekly total by over $1,100, resulting in a monthly estimate that's too low by more than $4,000. Choice C ($17,000 × 4 = $68,000) overestimates by rounding the weekly total up nearly $900, creating an inflated monthly figure. Choice D ($18,000 × 4 = $72,000) represents the largest error, overestimating the weekly costs by almost $2,000 and the monthly total by over $7,000. For estimation problems on the Accuplacer, calculate the exact answer first when possible, then identify which rounded calculation produces the closest result. Don't automatically choose the middle option – the "most reasonable" estimate is the one that minimizes error, not necessarily the one that seems moderate.
11. A factory produces widgets in batches. Machine A produces 2,847 widgets per hour, Machine B produces 3,156 widgets per hour, and Machine C produces 1,923 widgets per hour. If all machines run for 7 hours, approximately how many widgets will be produced?
    Answer: 56,000 widgets This is a multi-step word problem that tests your ability to calculate total production across multiple machines and time periods, then use estimation skills to identify the closest answer. First, find each machine's total production over 7 hours by multiplying hourly rates by time: Machine A produces $2,847 × 7 = 19,929$ widgets, Machine B produces $3,156 × 7 = 22,092$ widgets, and Machine C produces $1,923 × 7 = 13,461$ widgets. Adding these together: $19,929 + 22,092 + 13,461 = 55,482$ total widgets. Since the question asks for an approximation, you need to round to the nearest thousand. 55,482 rounds to approximately 55,000, making answer choice D) 56,000 widgets the closest match. Looking at the wrong answers: A) 48,000 widgets is too low—this might result from calculation errors or forgetting to include one machine's production. B) 54,000 widgets is close but still about 1,500 widgets short of the actual total. C) 62,000 widgets significantly overshoots the mark by about 6,500 widgets, possibly from adding incorrectly or miscalculating one of the multiplication steps. When tackling multi-step production problems like this, work systematically: calculate each machine's output separately, add them together, then round appropriately. Double-check your multiplication since small errors get magnified when you're dealing with large numbers and multiple machines.
12. The estimated answer to a multiplication problem is 3,200. Which of the following calculations was most likely being estimated?
    Answer: $42 \times 78$ Estimate the product for each option. A) $38 \times 41 \approx 40 \times 40 = 1,600$. B) $42 \times 78 \approx 40 \times 80 = 3,200$. C) $59 \times 63 \approx 60 \times 60 = 3,600$. D) $71 \times 89 \approx 70 \times 90 = 6,300$. The calculation that results in an estimate of 3,200 is $42 \times 78$.
13. A restaurant served 2,912 customers in a 7-day week. On average, approximately how many customers were served each day?
    Answer: 400 To find the approximate daily average, divide the total customers by the number of days. It is difficult to divide 2,912 by 7. Round 2,912 to a nearby number that is easily divisible by 7, such as 2,800. The estimated average is $2,800 \div 7 = 400$. The actual average is $2,912 \div 7 \approx 416$, so 400 is a very good estimate.
14. A school district is calculating approximate transportation costs. If 1,847 students need bus service at $127 per student per month for 9 months, which calculation provides the best estimate?
    Answer: $1,800 × $130 × 9 = $2,106,000 When you encounter estimation problems, you need to round the given numbers to make calculations easier while staying as close as possible to the original values. The goal is finding the most reasonable approximation, not the exact answer. Let's round each value strategically: 1,847 students rounds to either 1,800 or 2,000, $127 per student rounds to either $125 or $130, and 9 months stays at 9 (though it could round to 10). The key is choosing rounded values that balance simplicity with accuracy. Answer choice D ($1,800 × $130 × 9 = $2,106,000) provides the best estimate because it rounds 1,847 down to 1,800 (a difference of only 47) and $127 up to $130 (a difference of only $3), while keeping 9 months exact. This combination of rounding up and down helps balance out estimation errors. Answer A rounds too aggressively: $100 is $27 below the actual cost, making this significantly underestimated. Answer B compounds errors by rounding students up to 2,000, cost up to $130, AND months up to 10 – three upward rounds create a major overestimate. Answer C uses reasonable rounding for students (2,000) and cost ($125), but 2,000 students is 153 more than actual, while $125 is $2 less than actual, making the student count error more significant. For estimation problems on the Accuplacer, look for the answer choice that balances rounding up some numbers with rounding down others, and choose rounded values closest to the originals. Avoid options that round everything in the same direction.
15. A person earns $19 per hour and works 38 hours a week for 51 weeks a year. Which is the best estimate of this person's annual income?
    Answer: $38,000 To estimate the annual income, round the three values and multiply: $19/hour \approx $20/hour. 38 hours/week \approx 40 hours/week. 51 weeks/year \approx 50 weeks/year. Now multiply the estimates: $20 \times 40 \times 50$. First, $20 \times 40 = 800$. Then, $800 \times 50 = 40,000$. The closest answer choice is $38,000. The actual income is $19 \times 38 \times 51 = \$36,822$, so $38,000 is a more reasonable estimate than $40,000.
16. A concert venue is estimating ticket revenue. They have three seating sections: Section A has 1,247 seats at $89 each, Section B has 2,156 seats at $67 each, and Section C has 987 seats at $45 each. What is the best estimate of total revenue if the venue sells out?
    Answer: $310,000 Estimate each section: Section A: 1,247 ≈ 1,250 seats × $89 ≈ $90 = $112,500; Section B: 2,156 ≈ 2,200 seats × $67 ≈ $70 = $154,000; Section C: 987 ≈ 1,000 seats × $45 = $45,000. Total: $112,500 + $154,000 + $45,000 = $311,500 ≈ $310,000. Choice A significantly underestimates all sections. Choice B underestimates Section B's contribution. Choice D overestimates by rounding prices up too much.
17. A school is planning a field trip for 489 students. The cost for each student is $28 for the bus and $19 for admission. Which of the following is the most reasonable estimate for the total cost of the field trip?
    Answer: $25,000 First, estimate the cost per student by rounding: $28 is about $30, and $19 is about $20. The total estimated cost per student is $30 + $20 = $50. Next, round the number of students from 489 to 500. Finally, multiply the estimated number of students by the estimated cost per student: $500 \times \$50 = \$25,000$.
18. Which calculation produces the best estimate for the value of $813 \times 58$?
    Answer: $800 \times 60$ The exact value of $813 \times 58$ is 47,154. We need to find which estimation is closest. A) $800 \times 50 = 40,000$. B) $800 \times 60 = 48,000$. C) $810 \times 50 = 40,500$. D) $810 \times 60 = 48,600$. Comparing the results to the actual value, 48,000 is the closest estimate. This shows that rounding to the nearest hundred for 813 and nearest ten for 58 provides a better result than other rounding combinations.
19. A warehouse receives a shipment of 198 crates, and each crate contains 52 boxes. If 1,120 boxes are immediately shipped out, what is the best estimate of the number of boxes remaining in the warehouse?
    Answer: 8,900 This is a two-step problem. First, estimate the total number of boxes received. Round 198 crates up to 200 and 52 boxes down to 50. The estimated total is $200 \times 50 = 10,000$ boxes. Second, estimate the number of boxes shipped out by rounding 1,120 down to 1,100. Subtract the shipped boxes from the total: $10,000 - 1,100 = 8,900$.
20. A business has 18 employees, each working 38 hours a week at an average wage of $21 per hour. Which of the following is the most reasonable estimate for the total weekly payroll?
    Answer: $14,000 To estimate the payroll, multiply the three numbers. A good strategy is to round some numbers up and some down to compensate. Round 18 employees up to 20, 38 hours up to 40, and $21 down to $20. The estimate is $20 \times 40 \times 20 = 16,000$. Another good strategy is to round 38 to 40 and 21 to 20, and keep 18: $18 \times 40 \times 20 = 720 \times 20 = 14,400$. Given the choices, $14,000 is the most reasonable estimate. The actual amount is $18 \times 38 \times 21 = \$14,364$.

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