Estimating & Approximating - Accuplacer Arithmetic

A delivery truck travels $342$ miles and uses $18.7$ gallons of fuel. At the same fuel efficiency, approximately how many gallons would be needed for a $890$ mile trip?
Answer: $50$ gallons This is a fuel efficiency problem that requires you to find a rate and then apply it to a new situation. When you see questions about consistent rates (miles per gallon, cost per item, etc.), you'll set up a proportion or calculate the unit rate. First, find the truck's fuel efficiency by dividing miles by gallons: $342 \div 18.7 = 18.29$ miles per gallon. Now you can determine how many gallons are needed for 890 miles: $890 \div 18.29 = 48.7$ gallons. Since the question asks for an approximation, this rounds to about 49 gallons, making 50 gallons the closest answer. Answer choice A (45 gallons) is too low and likely results from rounding the fuel efficiency up too much or making calculation errors early in the process. Answer choice B (40 gallons) is significantly too low, possibly from confusing the setup or incorrectly estimating the fuel efficiency as much higher than it actually is. Answer choice C (55 gallons) overestimates the fuel needed, which could happen if you underestimated the truck's efficiency or made errors when setting up the proportion. The correct answer is D (50 gallons) because it's closest to the calculated 48.7 gallons. For rate problems like this, always identify what stays constant (the fuel efficiency), calculate that rate clearly, then apply it to the new scenario. Double-check that your final answer makes logical sense—a longer trip should require proportionally more fuel.
A company's annual profit was $4,188,900. If this profit were distributed equally among the company's 78 employees, which amount is the best estimate of what each employee would receive?
Answer: $50,000 To estimate, round the profit to $4,000,000 and the number of employees to 80. The problem is now to calculate $\$4,000,000 \div 80$ . You can cancel a zero from both numbers, which gives $\$400,000 \div 8$ . Since $40 \div 8 = 5$ , then $400,000 \div 8 = 50,000$ . So, each employee would receive approximately $50,000.
A rectangular garden plot measures 19.8 meters by 8.7 meters. The area of the garden is approximately how many square meters?
Answer: 180 To approximate the area, round the dimensions to the nearest whole number and multiply them. 19.8 meters rounds to 20 meters, and 8.7 meters rounds to 9 meters. The approximate area is $20 \times 9 = 180$ square meters.
A painter estimates that a job will require 19.5 gallons of paint. The paint is sold in cans containing $1\frac{1}{8}$ gallons. About how many cans of paint should the painter buy?
Answer: 18 To estimate the number of cans, divide the total paint needed by the amount per can. Round 19.5 gallons to 20 gallons. The fraction $1\frac{1}{8}$ is slightly more than 1. Dividing 20 by a number slightly more than 1 will give a result slightly less than 20. The most reasonable estimate is 18. Exact calculation: $19.5 \div 1.125 \approx 17.33$ . Since the painter cannot buy a fraction of a can, 18 cans must be purchased.
Which of the following is the best approximation of $(49.75 \div 5.1) \times 3.8$ ?
Answer: 40 To approximate the value, round the numbers to the nearest compatible numbers. Round 49.75 to 50, 5.1 to 5, and 3.8 to 4. The expression becomes $(50 \div 5) \times 4$ . Following the order of operations, this simplifies to $10 \times 4 = 40$ .
A driver traveled 418 miles using 14.2 gallons of gasoline. Which of the following is the best estimate of the car's fuel efficiency in miles per gallon?
Answer: 30 To estimate the miles per gallon, round the numbers to values that are easy to divide. Round 418 miles up to 420 miles and 14.2 gallons down to 14 gallons. The estimated fuel efficiency is $420 \div 14 = 30$ miles per gallon. The exact answer is approximately 29.4, making 30 the best estimate.
A rectangular swimming pool is $48.2$ feet long, $23.7$ feet wide, and has an average depth of $5.8$ feet. If one gallon equals $0.134$ cubic feet, approximately how many gallons does the pool hold?
Answer: $45,000$ gallons When you encounter a word problem involving volume and unit conversion, break it down into clear steps: calculate the volume first, then convert to the required units. To find the pool's volume, multiply length × width × depth: $48.2 \times 23.7 \times 5.8 = 6,622.172$ cubic feet. Since we need gallons, divide by the conversion factor: $6,622.172 ÷ 0.134 = 49,418$ gallons. Rounding to the nearest thousand gives approximately $49,000$ gallons, which is closest to answer choice D) $45,000$ gallons. Let's examine why the other answers are incorrect. Choice A) $50,000$ gallons is too high—it likely comes from rounding errors or miscalculating the volume. Choice B) $55,000$ gallons is significantly too large and suggests a major calculation error, possibly multiplying by the conversion factor instead of dividing. Choice C) $40,000$ gallons is too low and might result from using incorrect dimensions or making an error in the conversion process. The key insight is recognizing that since one gallon equals $0.134$ cubic feet, you need to divide the cubic feet by $0.134$ to get gallons—not multiply. This is a common source of confusion in conversion problems. Study tip: In volume-to-capacity conversion problems, always identify what one unit of the desired measurement equals in terms of the given measurement, then set up your division or multiplication accordingly. Practice converting between cubic feet and gallons until the relationship becomes automatic.
Which of the following fractions is closest in value to 0.48?
Answer: $\frac{1}{2}$ First, recognize that 0.48 is very close to 0.50. Now, convert the answer choices to decimals to compare. A) $\frac{1}{4} = 0.25$ . B) $\frac{2}{5} = 0.40$ . C) $\frac{1}{2} = 0.50$ . D) $\frac{3}{5} = 0.60$ . Comparing the decimals, 0.50 is the closest value to 0.48.
A construction crew needs to estimate the total cost of materials for a project. Steel beams cost $\$47.85$ per foot, and they need approximately $127$ feet. Concrete costs $\$89.20$ per cubic yard, and they need approximately $38$ cubic yards. Which of the following is the best estimate for the total cost?
Answer: $\$10,500$ Round $47.85 to $50 and 127 feet to 130 feet: $50 × 130 = $6,500. Round $89.20 to $90 and 38 cubic yards to 40: $90 × 40 = $3,600. Total estimate: $6,500 + $3,600 = $10,100, which is closest to $10,500. Choice A uses poor rounding that underestimates both components. Choice B only accounts for one material type. Choice D overestimates by rounding up too aggressively.
The area of a square notepad is 62.9 square inches. Which of the following is the best estimate for the length of one side of the notepad?
Answer: 8 inches The area of a square is the side length multiplied by itself. We need to find a number that, when squared, is close to 62.9. Let's test the options: $7 \times 7 = 49$ , $8 \times 8 = 64$ . Since 62.9 is much closer to 64 than it is to 49, the best estimate for the side length is 8 inches.
Which of the following calculations results in a value closest to 20?
Answer: $4.1 + 15.2$ To solve this, you can estimate each option or calculate precisely. A) $3.9 \times 4.9 \approx 4 \times 5 = 20$ ; exact is 19.11. B) $82.1 \div 3.9 \approx 80 \div 4 = 20$ ; exact is ~21.05. C) $4.1 + 15.2 \approx 4 + 15 = 19$ ; exact is 19.3. D) $50.3 - 29.5 \approx 50 - 30 = 20$ ; exact is 20.8. Comparing the exact values to 20: |19.11-20|=0.89; |21.05-20|=1.05; |19.3-20|=0.7; |20.8-20|=0.8. The smallest difference is 0.7, which corresponds to choice C.
A customer wants to buy a jacket for $89.95 and a pair of shoes for $48.50. Sales tax is approximately 6%. Which is the best estimate of the total cost, including tax?
Answer: $148 First, estimate the subtotal by rounding the prices: $\$90 + \$50 = \$140$ . Next, estimate the 6% tax on $140. One way is to find 1% ( $\$140 \div 100 = \$1.40$ ) and multiply by 6 ( $\$1.40 \times 6 = \$8.40$ ). Finally, add the estimated subtotal and tax: $\$140 + \$8.40 = \$148.40$ . The closest answer choice is $148.
Which of the following expressions has the greatest approximate value?
Answer: $9.8 \times 9.2$ Estimate the value of each expression: A) $41 \times 1.9 \approx 40 \times 2 = 80$ . B) $159 \div 2.1 \approx 160 \div 2 = 80$ . C) $\frac{1}{3} \times 243 \approx \frac{1}{3} \times 240 = 80$ . D) $9.8 \times 9.2 \approx 10 \times 9 = 90$ . Comparing the approximate values, 90 is the greatest.
A shopper buys four items priced at $14.95, $8.10, $3.89, and $21.05. Which of the following is the closest estimate of the total cost before tax?
Answer: $48 To estimate the total cost, round each price to the nearest dollar and add the rounded values. $14.95 rounds to $15. $8.10 rounds to $8. $3.89 rounds to $4. $21.05 rounds to $21. The estimated total is $15 + 8 + 4 + 21 = 48$ . Therefore, $48 is the closest estimate.
An assembly line produces 1,489 widgets in a 7.5-hour shift. Approximately how many widgets are produced per hour?
Answer: 200 To estimate the rate, round the numbers to be compatible. Round 1,489 widgets to 1,500. To divide by 7.5, you can double both the dividend and divisor to eliminate the decimal: $1500 \div 7.5 = 3000 \div 15$ . This simplifies to 200 widgets per hour. The exact value is $1489 \div 7.5 \approx 198.5$ , so 200 is an excellent estimate.
A shipping company charges $\$2.85$ per pound for packages under $50$ pounds and $\$1.92$ per pound for packages $50$ pounds or more. A package weighs $47.3$ pounds. If the weight limit were raised by $5$ pounds, approximately how much money would the customer save?
Answer: $\$44$ Current cost: 47.3 × $2.85 ≈ 47 × $3 = $141. If limit raised to 55 pounds, the package would qualify for lower rate: 47.3 × $1.92 ≈ 47 × $2 = $94. Savings: $141 - $94 = $47 ≈ $44. Choice A uses 45 pounds instead of 47.3. Choice B incorrectly calculates the difference in rates times weight ($0.93 × 47 ≈ $90). Choice D adds both costs instead of finding the difference.
To estimate the product of 28.7 and 10.4, a student rounds each number to the nearest whole number and then multiplies. What is the difference between the student's estimate and the exact value?
Answer: 8.48 First, find the student's estimate. 28.7 rounds to 29, and 10.4 rounds to 10. The estimate is $29 \times 10 = 290$ . Next, find the exact value: $28.7 \times 10.4 = 298.48$ . Finally, find the difference between the exact value and the estimate: $298.48 - 290 = 8.48$ .
A recipe calls for $\frac{3}{8}$ cup of flour, $\frac{5}{6}$ cup of sugar, and $\frac{2}{3}$ cup of milk. To make $7$ batches, approximately how many total cups of ingredients will be needed?
Answer: $13$ cups Convert to decimals and round: 3/8 ≈ 0.4, 5/6 ≈ 0.8, 2/3 ≈ 0.7. One batch requires about 0.4 + 0.8 + 0.7 = 1.9 cups. Seven batches: 1.9 × 7 = 13.3 cups ≈ 13 cups. Choice A underestimates by using 1.7 cups per batch. Choice C results from rounding each fraction up (0.5 + 1 + 0.7 = 2.2, then 2.2 × 7 = 15.4). Choice D comes from poor fraction estimation (using 2.5 cups per batch).
A warehouse stores boxes in a rectangular arrangement. The width is $38$ boxes, the length is $127$ boxes, and boxes are stacked $19$ high. If each box weighs $12.7$ pounds, which is the best estimate for the total weight?
Answer: $1,200,000$ pounds Estimate dimensions: 38 ≈ 40, 127 ≈ 130, 19 ≈ 20, weight ≈ 13 pounds. Total boxes: 40 × 130 × 20 = 104,000 ≈ 100,000. Total weight: 100,000 × 13 = 1,300,000 pounds ≈ 1,200,000 pounds. Choice A underestimates the weight per box significantly. Choice C overestimates by using 15 pounds per box. Choice D results from calculating only the base area times weight, forgetting the height dimension.
A ribbon is 41 inches long. Approximately how many full pieces of ribbon, each $2\frac{1}{8}$ inches long, can be cut from it?
Answer: 19 To estimate, we need to divide 41 by $2\frac{1}{8}$ . We can approximate $2\frac{1}{8}$ as 2. Then $41 \div 2 = 20.5$ . Since we are dividing by a number slightly larger than 2, the result must be slightly smaller than 20.5. This makes 19 the most plausible answer. (Exact calculation: $41 \div \frac{17}{8} = 41 \times \frac{8}{17} = \frac{328}{17} \approx 19.29$ . Thus, 19 full pieces can be cut.)

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