Apply Dimensional Analysis - TEAS Mathematics

1. A factory machine produces plastic sheeting that is 150 centimeters wide. The machine produces the sheeting at a rate of 30 meters per minute. The plastic has a thickness of 2 millimeters. What is the volume of plastic produced in one 8-hour shift, in cubic meters?
   Answer: 43.2 m³ This problem requires calculating a volume from a rate of area production and a thickness. First, convert all measurements to meters. Width = 150 cm = 1.5 m. Thickness = 2 mm = 0.002 m. The production rate is 30 m/min. In an 8-hour shift, the total length produced is: Length = (30 m/min) × (60 min/hr) × 8 hr = 14,400 m. Now, calculate the total volume by multiplying length, width, and thickness: Volume = 14,400 m × 1.5 m × 0.002 m = 43.2 m³. The total volume produced is 43.2 cubic meters.
2. A medication is dosed at 15 mg per kilogram of body weight, administered every 8 hours. Using the conversion chart shown, what is the total dose in grams administered over a 3-day period to a patient weighing 176 pounds?
   Answer: $10.8$ g Weight: $176 \text{ lb} \times \frac{1 \text{ kg}}{2.2 \text{ lb}} = 80$ kg. Single dose: $80 \times 15 = 1200$ mg = 1.2 g. Doses in 3 days: $3 \times 24 / 8 = 9$ doses. Total: $9 \times 1.2 = 10.8$ g. Choice B uses only 3 doses (one per day). Choice C multiplies by 27 hours instead of 9 doses. Choice D gives a single dose.
3. A land surveyor measures a rectangular plot of land to be 350 rods long and 80 rods wide. An older deed lists the area of this plot in hectares. What is the area of the plot in hectares? (Use 1 rod = 16.5 feet, 1 acre = 43,560 square feet, and 1 hectare = 2.47 acres).
   Answer: 70.0 ha First, calculate the area of the plot in square rods: Area = 350 rods × 80 rods = 28,000 square rods. Next, convert square rods to square feet. Since 1 rod = 16.5 feet, then 1 square rod = (16.5 ft)² = 272.25 ft². So, Area = 28,000 rod² × 272.25 ft²/rod² = 7,623,000 ft². Now, convert the area from square feet to acres: Area = 7,623,000 ft² ÷ 43,560 ft²/acre = 175 acres. Finally, convert the area from acres to hectares: Area = 175 acres ÷ 2.47 acres/hectare ≈ 70.85 hectares. The closest answer is 70.0 ha.
4. Convert $2{,}500$ square centimeters to square meters.
   Answer: $0.25\,\text{m}^2$ When converting between area units, you need to square the linear conversion factor. This is a crucial distinction from simple length conversions that often trips students up on the TEAS. To convert from square centimeters to square meters, start with the basic length conversion: $1 \text{ meter} = 100 \text{ centimeters}$. This means $1 \text{ centimeter} = 0.01 \text{ meters}$. For area conversions, you must square this factor: $(0.01)^2 = 0.0001$. Therefore, $1 \text{ cm}^2 = 0.0001 \text{ m}^2$. Now convert: $2{,}500 \text{ cm}^2 \times 0.0001 \frac{\text{m}^2}{\text{cm}^2} = 0.25 \text{ m}^2$ Answer B ($0.25 \text{ m}^2$) is correct. Answer A ($0.025 \text{ m}^2$) results from incorrectly cubing the conversion factor instead of squaring it. Answer C ($2.5 \text{ m}^2$) comes from using the linear conversion factor ($0.01$) directly without squaring it. Answer D ($25 \text{ m}^2$) occurs when students multiply by $100$ instead of dividing, forgetting that centimeters are smaller units than meters. Study tip: Remember the pattern for area conversions: always square the linear conversion factor. Since there are $100$ cm in $1$ m, there are $100^2 = 10{,}000$ cm² in $1$ m². When you see area unit conversions on the TEAS, immediately think "square the factor" to avoid the most common mistake.
5. The table provides production data for a factory. If the factory runs 16 hours per day, 6 days per week, what mass of product, in metric tons, is produced in 4 weeks? (1 metric ton = 1000 kg)
   Answer: $57.6$ metric tons $150 \frac{\text{kg}}{\text{hr}} \times 16 \frac{\text{hr}}{\text{day}} \times 6 \frac{\text{day}}{\text{wk}} \times 4 \text{ wk} = 57{,}600$ kg = 57.6 metric tons. Choice B uses 1 week instead of 4. Choice C divides by 10,000 instead of 1000. Choice D forgets to divide by 1000 for the final conversion.
6. Refer to the recipe card shown. A commercial kitchen needs to scale the recipe from 8 servings to 150 servings. How many quarts of milk will be needed?
   Answer: $14.06$ qt Per serving: $\frac{3 \text{ cups}}{8} = 0.375$ cups. For 150: $150 \times 0.375 = 56.25$ cups. Fluid oz: $56.25 \times 8 = 450$ fl oz. Quarts: $450/32 \approx 14.06$ qt. Choice A divides cups by 32 directly (forgetting cup-to-oz). Choice B uses 16 oz/pint confusion. Choice D multiplies instead of dividing at the last step.
7. An agricultural field yields 180 bushels of corn per acre. Each bushel of corn weighs approximately 56 pounds. If a farmer sells the corn for $0.25 per kilogram, what is the revenue generated per hectare of this field? (Use 1 hectare ≈ 2.47 acres, and 1 kilogram ≈ 2.205 pounds).
   Answer: $2,788 This is a multi-step conversion of a rate. Start with the yield in bushels per acre and convert it to dollars per hectare. First find the mass in pounds per acre: 180 bushels/acre × 56 lb/bushel = 10,080 lb/acre. Next, convert this mass to kilograms per acre: 10,080 lb/acre ÷ 2.205 lb/kg ≈ 4571.4 kg/acre. Now, calculate the revenue in dollars per acre: 4571.4 kg/acre × $0.25/kg = $1142.85/acre. Finally, convert this revenue rate from per acre to per hectare: $1142.85/acre × 2.47 acres/hectare ≈ $2822.84/hectare. The closest answer choice is $2,788.
8. Convert $48$ ounces to gallons. (Use $1\,\text{gallon}=128\,\text{oz}$.)
   Answer: $0.375\,\text{gal}$ Unit conversion problems require you to set up a proportion or use dimensional analysis to change from one unit to another. When converting between units, you need to determine whether you're going from a smaller unit to a larger unit (which means dividing) or from a larger unit to a smaller unit (which means multiplying). To convert 48 ounces to gallons, you're going from a smaller unit (ounces) to a larger unit (gallons), so you'll divide. Using the given conversion factor of 1 gallon = 128 ounces, set up the calculation: $48 \text{ oz} \div 128 \text{ oz/gal} = \frac{48}{128} = 0.375 \text{ gallons}$. You can simplify this fraction by dividing both numerator and denominator by 16: $\frac{48}{128} = \frac{3}{8} = 0.375$. Answer A (0.125 gal) represents $\frac{1}{8}$ of a gallon, which would be 16 ounces—this comes from incorrectly dividing 128 by 8 instead of finding what fraction 48 is of 128. Answer B (0.250 gal) equals $\frac{1}{4}$ gallon or 32 ounces—this might result from confusing 48 with 32 or miscalculating the fraction. Answer D (0.750 gal) equals $\frac{3}{4}$ gallon or 96 ounces—this could come from doubling the correct answer or making an arithmetic error. For unit conversions on the TEAS, always write down the conversion factor clearly and double-check whether you should multiply or divide. When going to a larger unit, the number gets smaller, and vice versa.
9. Convert $72$ inches to meters. (Use $1\,\text{in}=2.54\,\text{cm}$.)
   Answer: $1.83\,\text{m}$ Unit conversion problems require you to set up a chain of conversions that cancels out unwanted units and leaves you with the desired units. When converting between measurement systems like inches to meters, you'll often need multiple conversion steps. To convert 72 inches to meters, you need two conversions: inches to centimeters, then centimeters to meters. Start with the given conversion: $1 \text{ in} = 2.54 \text{ cm}$. Set up your calculation so units cancel properly: $72 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}} \times \frac{1 \text{ m}}{100 \text{ cm}} = \frac{72 \times 2.54}{100} \text{ m} = \frac{182.88}{100} \text{ m} = 1.8288 \text{ m}$ Rounded to two decimal places, this gives $1.83 \text{ m}$, which is answer choice B. Looking at the wrong answers: Choice A ($0.183 \text{ m}$) results from dividing by 10 instead of 100 when converting centimeters to meters—a common decimal place error. Choice C ($18.3 \text{ m}$) comes from multiplying by 10 instead of dividing by 100, while choice D ($183 \text{ m}$) represents forgetting the centimeter-to-meter conversion entirely, leaving you with the centimeter value as if it were meters. For TEAS unit conversion questions, always write out your conversion factors as fractions and verify that units cancel properly. Double-check your decimal placement—most errors in metric conversions involve moving the decimal point the wrong direction or wrong number of places.
10. A recipe calls for $20$ fluid ounces of milk. How many milliliters of milk is this? (Use $1\,\text{fl oz}=29.573\,\text{mL}$.)
    Answer: $591\,\text{mL}$ Unit conversion questions test your ability to multiply by the correct conversion factor to change from one measurement system to another. When converting from fluid ounces to milliliters, you need to multiply by the given conversion rate. To find how many milliliters are in 20 fluid ounces, multiply the volume by the conversion factor: $20 \text{ fl oz} \times 29.573 \frac{\text{mL}}{\text{fl oz}} = 591.46 \text{ mL}$. Rounding to the nearest whole number gives you 591 mL, making C the correct answer. Let's examine why the other options are wrong. Choice A (296 mL) represents roughly half the correct answer—this could result from accidentally dividing 591 by 2 or misreading the problem. Choice B (473 mL) is close to what you'd get if you used an incorrect conversion factor of about 23.65 mL per fluid ounce, which might come from confusing different unit conversions. Choice D (750 mL) would require a conversion factor of 37.5 mL per fluid ounce, which is significantly higher than the actual rate and might result from rounding errors or using an incorrect conversion altogether. For TEAS unit conversion problems, always write out your calculation with units included—this helps you verify that you're multiplying (not dividing) by the conversion factor. Double-check that your final answer makes sense: since 1 fluid ounce equals about 30 mL, 20 fluid ounces should equal roughly 600 mL, which matches our calculated result.
11. A rectangular swimming pool is 50 feet long, 25 feet wide, and has an average depth of 6 feet. The pool is being filled at a rate of 200 gallons per minute. How long will it take to fill the pool to 80% of its capacity, in hours? (Use the conversion 1 cubic foot ≈ 7.48 gallons)
    Answer: 3.7 hours First, calculate the total volume of the pool in cubic feet: V = 50 ft × 25 ft × 6 ft = 7,500 ft³. Next, determine the volume to be filled, which is 80% of the total: 7,500 ft³ × 0.80 = 6,000 ft³. Convert this volume to gallons: 6,000 ft³ × 7.48 gal/ft³ = 44,880 gallons. Now, calculate the time in minutes to fill this volume at the given rate: 44,880 gal ÷ 200 gal/min = 224.4 minutes. Finally, convert the time from minutes to hours: 224.4 min ÷ 60 min/hr ≈ 3.74 hours. The closest answer is 3.7 hours.
12. How many tons (short tons) are equivalent to $8{,}000$ pounds? (Use $1\,\text{ton}=2{,}000\,\text{lb}$.)
    Answer: $4\,\text{tons}$ Unit conversion problems like this test your ability to use conversion factors systematically. When you see different units of measurement, set up a conversion that cancels out the unwanted unit and leaves you with the desired unit. To convert $8{,}000$ pounds to tons, you need to use the given conversion factor: $1\,\text{ton} = 2{,}000\,\text{lb}$. Set up your conversion so the pounds cancel out: $8{,}000\,\text{lb} \times \frac{1\,\text{ton}}{2{,}000\,\text{lb}} = \frac{8{,}000}{2{,}000}\,\text{tons} = 4\,\text{tons}$ The pounds units cancel, leaving you with tons. Dividing $8{,}000$ by $2{,}000$ gives you $4$. Looking at the incorrect answers: Choice A ($1\,\text{ton}$) would result from incorrectly dividing $2{,}000$ by $8{,}000$ instead of the other way around. Choice B ($2\,\text{tons}$) might come from dividing $8{,}000$ by $4{,}000$ instead of $2{,}000$, perhaps confusing the conversion factor. Choice C ($3\,\text{tons}$) doesn't match any logical calculation error but could result from misremembering the conversion factor or arithmetic mistakes. For unit conversion success on the TEAS, always write out your conversion factor as a fraction that cancels the unwanted unit. Check that your answer makes sense — since tons are larger units than pounds, you should expect fewer tons than pounds, which $4$ tons versus $8{,}000$ pounds confirms.
13. If a runner completes a $45$-kilometer race, how many miles has the runner covered? (Use $1\,\text{mile}=1.609\,\text{km}$.)
    Answer: $28.0\,\text{mi}$ This question tests unit conversion, a fundamental skill in healthcare where you'll frequently convert between metric and imperial measurements. When converting units, you need to set up the conversion factor so that unwanted units cancel out. To convert 45 kilometers to miles, you'll use the given conversion factor: $1\,\text{mile} = 1.609\,\text{km}$. Set up your calculation as: $45\,\text{km} \times \frac{1\,\text{mile}}{1.609\,\text{km}} = \frac{45}{1.609}\,\text{miles} = 27.97\,\text{miles}$, which rounds to $28.0\,\text{miles}$. Notice how the kilometers cancel out, leaving you with miles. Choice A ($18.4\,\text{mi}$) represents a calculation error—you might get this if you incorrectly divided by a larger number or made an arithmetic mistake. Choice C ($38.6\,\text{mi}$) suggests you may have set up the conversion incorrectly or used the wrong conversion factor entirely. Choice D ($72.2\,\text{mi}$) is what you'd get if you multiplied instead of divided: $45 \times 1.609 = 72.405$. This is a common trap—remember that since kilometers are larger units than miles, you should expect fewer miles than kilometers. For TEAS unit conversion problems, always write out your setup completely with units included. This helps you catch errors before calculating. Remember: when converting from a larger unit to a smaller unit, your final number should be bigger; when converting from smaller to larger (like km to miles), your answer should be smaller than what you started with.
14. A pump delivers $3$ gallons per minute. What is this flow rate in liters per hour? (Use $1\,\text{gal}=3.785\,\text{L}$.)
    Answer: $681\,\dfrac{\text{L}}{\text{h}}$ When you encounter unit conversion problems involving rates, you need to convert both parts of the rate separately - in this case, gallons to liters and minutes to hours. Start with the given rate: $3$ gallons per minute. To convert gallons to liters, multiply by the conversion factor: $3 \text{ gal} \times 3.785 \frac{\text{L}}{\text{gal}} = 11.355 \text{ L}$. So the pump delivers $11.355$ liters per minute. Next, convert minutes to hours. Since there are $60$ minutes in an hour, multiply by $60$: $11.355 \frac{\text{L}}{\text{min}} \times 60 \frac{\text{min}}{\text{h}} = 681.3 \frac{\text{L}}{\text{h}}$. This rounds to $681 \frac{\text{L}}{\text{h}}$, making answer D correct. Looking at the wrong answers: Choice A ($11.4 \frac{\text{L}}{\text{h}}$) represents the trap of converting gallons to liters but forgetting to convert minutes to hours - you'd get this if you stopped after finding $11.355$ liters per minute. Choice B ($68.1 \frac{\text{L}}{\text{h}}$) suggests an error in the time conversion, perhaps dividing by $10$ instead of multiplying by $60$. Choice C ($181 \frac{\text{L}}{\text{h}}$) likely results from calculation errors in either the unit conversion or arithmetic. Remember: when converting compound units like rates, convert each unit separately and keep track of whether you're multiplying or dividing. Always double-check that your final units match what the question asks for.
15. A water treatment plant processes 5 million gallons of water per day. The water contains a contaminant at a concentration of 2.5 parts per million (ppm) by mass. The plant must dose the water with a chemical that neutralizes the contaminant at a ratio of 10 grams of chemical for every 1 gram of contaminant. How many kilograms of the chemical are needed per week? (Use 1 gallon of water ≈ 3.785 kg and 1 million = 10⁶).
    Answer: 3,312 kg First, find the total mass of water processed per week. Mass per day = 5,000,000 gal × 3.785 kg/gal = 18,925,000 kg. Mass per week = 18,925,000 kg/day × 7 days/week = 132,475,000 kg/week. Next, find the total mass of the contaminant. A concentration of 2.5 ppm means there are 2.5 kg of contaminant for every 10⁶ kg of water. Mass of contaminant = (132,475,000 kg water) × (2.5 kg contaminant / 10⁶ kg water) ≈ 331.19 kg of contaminant per week. Now, determine the mass of the neutralizing chemical needed using the 10:1 mass ratio: 331.19 kg contaminant × 10 = 3311.9 kg of chemical. The closest answer is 3,312 kg.
16. Express $120$ grams as ounces. (Use $1\,\text{oz}=28.3495\,\text{g}$.)
    Answer: $4.23\,\text{oz}$ Unit conversion problems require you to set up a proportion or use dimensional analysis to convert between different measurement systems. When converting from grams to ounces, you need to divide by the conversion factor since you're going from a smaller unit to a larger one. To convert 120 grams to ounces, use the given conversion factor: $1\,\text{oz} = 28.3495\,\text{g}$. Set up the calculation as: $120\,\text{g} \times \frac{1\,\text{oz}}{28.3495\,\text{g}} = \frac{120}{28.3495} = 4.23\,\text{oz}$. The grams cancel out, leaving you with ounces. Choice A ($2.82\,\text{oz}$) represents a calculation error, possibly from incorrectly using a rounded conversion factor or making an arithmetic mistake. Choice C ($7.09\,\text{oz}$) suggests you might have used an incorrect conversion factor, perhaps confusing ounces with a different unit. Choice D ($28.3\,\text{oz}$) is clearly wrong—it's nearly equal to the conversion factor itself and would mean 120 grams equals almost a pound, which is far too much. Remember that grams are much smaller than ounces, so when converting from grams to ounces, your answer should be significantly smaller than the original number. A quick reasonableness check helps: since roughly 28 grams equal 1 ounce, 120 grams should be about 4 ounces. Always set up your conversion factor so the units you're converting from appear in the denominator and cancel out.
17. Convert $2.8$ meters to millimeters.
    Answer: $2{,}800\,\text{mm}$ When converting between metric units, you're working with powers of 10, which makes the process systematic once you understand the relationships. The key is knowing how the prefixes relate to the base unit. To convert from meters to millimeters, you need to understand that "milli" means one-thousandth, so 1 millimeter = 0.001 meters, or conversely, 1 meter = 1,000 millimeters. When converting from a larger unit to a smaller unit, you multiply by the conversion factor. Starting with 2.8 meters: $2.8 \text{ meters} \times \frac{1{,}000 \text{ mm}}{1 \text{ meter}} = 2{,}800 \text{ mm}$ You can also think of this as moving the decimal point three places to the right (since 1,000 has three zeros): 2.8 becomes 2,800. Looking at the wrong answers: Choice A ($2.8\,\text{mm}$) represents no conversion at all—just changing the unit label without adjusting the number. Choice B ($28\,\text{mm}$) results from multiplying by 10 instead of 1,000, confusing millimeters with centimeters. Choice C ($280\,\text{mm}$) comes from multiplying by 100, perhaps confusing the conversion with meters to centimeters. For metric conversions on the TEAS, memorize the key prefixes: kilo (1,000), centi (0.01), and milli (0.001). Remember that going from larger to smaller units means multiplying, while going from smaller to larger means dividing. The decimal point movement follows the number of zeros in the conversion factor.
18. Using the unit conversions shown in the diagram, convert a vehicle speed of 88 feet per second into kilometers per hour. (Use 1 mile = 1.609 km.)
    Answer: $96.5$ km/hr $88 \frac{\text{ft}}{\text{s}} \times \frac{1 \text{ mi}}{5280 \text{ ft}} \times \frac{3600 \text{ s}}{1 \text{ hr}} \times \frac{1.609 \text{ km}}{1 \text{ mi}} = 60 \times 1.609 \approx 96.5$ km/hr. Choice A stops at mph. Choice C converts ft to m only without the time conversion. Choice D incorrectly multiplies by 1.609 twice.
19. A European car has a fuel efficiency rating of 12.0 kilometers per liter. If gasoline costs $3.50 per gallon in the United States, what would be the cost of fuel for this car to travel 500 miles? (Use 1 mile ≈ 1.609 km and 1 gallon ≈ 3.785 L).
    Answer: $58.85 First, convert the travel distance from miles to kilometers: 500 miles × 1.609 km/mile = 804.5 km. Next, calculate the total liters of fuel required for the trip: 804.5 km ÷ 12.0 km/L = 67.04 L. Now, convert the required liters of fuel to gallons: 67.04 L ÷ 3.785 L/gallon ≈ 16.81 gallons. Finally, calculate the total cost of the fuel: 16.81 gallons × $3.50/gallon ≈ $58.84. The closest answer is $58.85.
20. Refer to the unit relationships in the diagram. A fuel pump delivers gasoline at 40 liters per minute. Expressed in cubic feet per hour, what is the flow rate? (1 L = 0.0353 ft³)
    Answer: $84.7$ ft³/hr $40 \frac{\text{L}}{\text{min}} \times \frac{60 \text{ min}}{1 \text{ hr}} \times \frac{0.0353 \text{ ft}^3}{1 \text{ L}} = 2400 \times 0.0353 \approx 84.7$ ft³/hr. Choice A converts without the time factor. Choice C fails to multiply 0.0353 (uses 1000 factor). Choice D divides 40 by something incorrect.

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