Unit Conversions - PSAT Math

1. The pressure on the ocean floor being studied is 3.2 atmospheres. What is this pressure in pascals? (1 atm = 101.3 kPa and 1 kPa = 1,000 Pa)
   Answer: $3.24 \times 10^{5}$ Pa Unit conversion problems require you to work systematically through multiple conversion factors, keeping track of units at each step to ensure they cancel properly. Starting with 3.2 atmospheres, you need to convert to pascals using the given conversion factors. First, convert atmospheres to kilopascals: $3.2 \text{ atm} \times \frac{101.3 \text{ kPa}}{1 \text{ atm}} = 324.16 \text{ kPa}$ Next, convert kilopascals to pascals: $324.16 \text{ kPa} \times \frac{1,000 \text{ Pa}}{1 \text{ kPa}} = 324,160 \text{ Pa}$ In scientific notation, this is $3.24 \times 10^5 \text{ Pa}$, which is answer choice B. Choice A ($3.24 \times 10^4 \text{ Pa}$) results from forgetting the final conversion from kilopascals to pascals—stopping at 324.16 kPa and incorrectly expressing this as pascals. Choice C ($3.24 \times 10^6 \text{ Pa}$) adds an extra factor of 10, perhaps from misreading one of the conversion factors. Choice D ($3.24 \times 10^7 \text{ Pa}$) compounds multiple errors, likely from both misreading conversion factors and calculation mistakes. When tackling unit conversions, always write out each step with units included, and verify that unwanted units cancel out completely. Double-check that your final answer makes sense—since pascals are much smaller units than atmospheres, you should expect a large number. Setting up conversion factors as fractions helps you see visually whether units will cancel properly.
2. A cyclist rides at a constant rate of 250 meters per minute for 18 minutes. How many kilometers does the cyclist travel? (Use $1000\text{ m}=1\text{ km}$.)
   Answer: 4.5 km We need to find the total distance traveled by a cyclist riding at 250 m/min for 18 minutes, expressed in kilometers. First, calculate total distance: 250 m/min × 18 min = 4,500 m. Then convert to kilometers: 4,500 m ÷ 1000 = 4.5 km. The dimensional analysis shows: (250 m/min) × (18 min) × (1 km/1000 m) = 4.5 km. A common error would be to convert the rate to km/min first, which adds unnecessary complexity. For distance = rate × time problems, calculate in the given units first, then convert the result.
3. A height of 5 feet 8 inches is measured for a basketball player. What is this height, to the nearest centimeter, if 1 inch equals 2.54 centimeters?
   Answer: 173 cm Unit conversion problems like this require careful step-by-step calculation and attention to detail. You need to convert mixed units (feet and inches) to a single unit, then apply the given conversion factor. First, convert the height to inches only. Since 5 feet 8 inches equals $5 \times 12 + 8 = 60 + 8 = 68$ inches, you're working with 68 total inches. Next, apply the conversion factor: 1 inch = 2.54 centimeters. Multiply the total inches by this factor: $68 \times 2.54 = 172.72$ centimeters. Finally, round to the nearest centimeter: 172.72 cm rounds to 173 cm, which is answer choice C. Looking at the wrong answers: Choice A (150 cm) is far too small—this would represent someone under 5 feet tall, suggesting a major calculation error or forgetting to include the feet measurement. Choice B (163 cm) is close but still too small; you might get this if you miscalculated the feet-to-inches conversion or made an arithmetic error in the multiplication. Choice D (187 cm) is too large, possibly resulting from adding the feet and inches incorrectly or using the wrong conversion factor. When tackling unit conversion problems, always convert to a single unit first before applying conversion factors. Double-check your arithmetic, especially when dealing with mixed measurements like feet and inches. These problems test both your calculation skills and your ability to work methodically through multiple steps without making careless errors.
4. A package weighs $3$ pounds $8$ ounces. How many ounces does it weigh? (Use $16\text{ oz}=1\text{ lb}$.)
   Answer: 56 oz We need to convert 3 pounds 8 ounces to total ounces using the conversion 16 oz = 1 lb. First, convert the pounds to ounces: 3 lb × 16 oz/lb = 48 oz. Then add the additional 8 ounces: 48 oz + 8 oz = 56 oz total. A common mistake is forgetting to add the extra 8 ounces and answering 48 oz, or miscalculating 3 × 16. When dealing with mixed units, convert the larger unit first, then add any remainder in the smaller unit.
5. A recipe needs $3.5$ quarts of broth. How many cups is this? Use $1\text{ qt}=2\text{ pt}$ and $1\text{ pt}=2\text{ cups}$.
   Answer: 14 cups The question asks how many cups are in 3.5 quarts of broth, using the conversions 1 quart = 2 pints and 1 pint = 2 cups. To convert quarts to cups, set up the conversion factors as 2 pints per quart and 2 cups per pint. Using dimensional analysis, start with 3.5 qt and multiply by (2 pt / 1 qt) to get pints, then multiply by (2 cups / 1 pt) to get cups, with units canceling: quarts cancel with quarts, pints cancel with pints, leaving cups. Calculate 3.5 × 2 = 7 pints, then 7 × 2 = 14 cups; alternatively, since 1 quart = 4 cups, directly 3.5 × 4 = 14 cups. A key error to avoid is halving instead of multiplying by the factors, which might lead to 7 cups or other fractions. Another mistake could be confusing pints and quarts in the chain. When taking tests, always verify the combined conversion factor (like 4 cups per quart) to simplify calculations and track units effectively.
6. A runner completes $5$ kilometers in $25$ minutes. What is the runner's average speed in meters per second? Use $1$ km $= 1000$ m and $60$ s $= 1$ min.
   Answer: 3.33 m/s The question asks for a runner's average speed in meters per second after completing 5 kilometers in 25 minutes, using 1 km = 1000 m and 60 s = 1 min. Set up the conversion with speed = distance/time, converting distance to meters and time to seconds using factors 1000 m/1 km and 60 s/1 min. First, convert distance: 5 km × (1000 m/1 km) = 5000 m; then time: 25 min × (60 s/1 min) = 1500 s; now speed = 5000 m / 1500 s = 3.333 m/s, with units canceling to m/s. Rounding to two decimals gives 3.33 m/s. A key error is converting time to seconds but forgetting to convert kilometers to meters, leading to 5 / 1500 = 0.0033 km/s. Another mistake is using minutes in the denominator without conversion, yielding km/min. In rate problems, perform all conversions before dividing to track units via dimensional analysis.
7. A package has a mass of $3.6$ kilograms. How many grams is this? Use $1\text{ kg}=1000\text{ g}$.
   Answer: 3600 g The question asks how many grams are in a package with a mass of 3.6 kilograms, using the conversion 1 kilogram = 1000 grams. Set up the conversion factor as 1000 grams per kilogram. Using dimensional analysis, start with 3.6 kg and multiply by (1000 g / 1 kg), canceling kilograms and leaving grams. The calculation is 3.6 × 1000 = 3600 g. Common errors include shifting the decimal incorrectly, leading to 36 g or 0.0036 g, or confusing kilo- with other prefixes. Another mistake is dividing instead of multiplying by 1000. A good strategy is to use dimensional analysis to track whether you're converting to smaller or larger units and verify with known equivalents.
8. A runner completes 800 meters in 2.5 minutes. What is the runner's speed in kilometers per hour? Use $1000\text{ m}=1\text{ km}$ and $60\text{ min}=1\text{ hr}$. Convert both distance and time units.
   Answer: 19.2 km/h This question requires finding a runner's speed in kilometers per hour from 800 meters in 2.5 minutes, using 1000 meters = 1 kilometer and 60 minutes = 1 hour. Set up the conversion: speed = 800 m / 2.5 min × (1 km / 1000 m) × (60 min / 1 hr) to cancel meters and minutes, leaving km/hr. First, 800 ÷ 2.5 = 320 m/min, then 320 × (60 / 1000) = 320 × 0.06 = 19.2 km/hr, with units canceling properly. Dimensional analysis is essential for rates, showing each factor's role in unit conversion. Common errors include forgetting the time conversion, yielding 0.32 km/min instead of per hour, or inverting factors to get 4.8 km/h. Another mistake is converting distance only, leading to 32 km in some miscalculation. As a strategy, compute speed in original units first, then apply conversions step-by-step, verifying units match the target.
9. A car's fuel economy is rated at 30 miles per gallon. What is the equivalent rating in kilometers per liter? (1 mile = 1.609 km and 1 gallon = 3.785 L)
   Answer: 12.8 km/L Unit conversion problems require you to set up conversion factors that cancel unwanted units while preserving the desired ones. When converting fuel economy from miles per gallon to kilometers per liter, you need to convert both the numerator (miles to kilometers) and denominator (gallons to liters). Start with 30 miles per gallon and set up your conversion factors: $30 \frac{\text{miles}}{\text{gallon}} \times \frac{1.609 \text{ km}}{1 \text{ mile}} \times \frac{1 \text{ gallon}}{3.785 \text{ L}}$ The "miles" units cancel, and the "gallon" units cancel, leaving kilometers per liter: $30 \times \frac{1.609}{3.785} = 30 \times 0.425 = 12.75 \text{ km/L}$ Rounding to one decimal place gives 12.8 km/L, which is answer choice C. Looking at the wrong answers: Choice A (8.5 km/L) likely results from incorrectly dividing instead of multiplying by one of the conversion factors. Choice B (10.5 km/L) suggests an error in the arithmetic or using approximate conversion factors incorrectly. Choice D (15.0 km/L) probably comes from flipping one of the conversion ratios, such as using 3.785 km per mile instead of 1.609. For unit conversion success on the PSAT, always write out your conversion factors as fractions and verify that unwanted units cancel before calculating. Double-check that your conversion factors are oriented correctly—the unit you want to eliminate should appear in both a numerator and denominator.
10. A factory produces $2.16\times10^{6}$ bolts each day. About how many bolts does the factory produce each second? (1 day = 86,400 seconds)
    Answer: 25 bolts per second This problem tests your ability to work with scientific notation and unit conversions—two essential skills that often appear together on the PSAT. To find bolts per second, you need to divide the daily production by the number of seconds in a day: $\frac{2.16 \times 10^6 \text{ bolts}}{86,400 \text{ seconds}}$. First, convert 86,400 to scientific notation: $86,400 = 8.64 \times 10^4$. Now you have: $\frac{2.16 \times 10^6}{8.64 \times 10^4}$ When dividing numbers in scientific notation, divide the coefficients and subtract the exponents: $\frac{2.16}{8.64} \times 10^{6-4} = 0.25 \times 10^2 = 25$ bolts per second. Choice A (2.5) likely results from an error with powers of 10—perhaps forgetting to convert 86,400 to scientific notation properly or miscalculating the exponent arithmetic. Choice C (250) suggests multiplying instead of dividing the exponents, giving $10^{6+4} = 10^{10}$ instead of $10^{6-4} = 10^2$. Choice D (2,500) combines multiple errors, possibly treating the division incorrectly and mishandling the scientific notation. The correct answer is B) 25 bolts per second. Strategy tip: When working with scientific notation division, remember the acronym "DSSM"—Divide coefficients, Subtract exponents, Simplify, Move decimal if needed. Always convert regular numbers to scientific notation first to avoid calculation errors with large numbers.
11. A household uses 500 kilowatt-hours (kWh) of electricity in a month. How much energy is this in joules? (1 kWh = $3.6\times10^{6}$ joules)
    Answer: $1.8\times10^{9}$ J This is a unit conversion problem that tests your ability to work with scientific notation and dimensional analysis. When you see energy units being converted, set up the conversion factor carefully and pay close attention to the powers of 10. To convert 500 kWh to joules, you multiply by the given conversion factor: $500 \text{ kWh} \times 3.6 \times 10^6 \text{ J/kWh}$. First, multiply the numbers: $500 \times 3.6 = 1800 = 1.8 \times 10^3$. Then handle the power of 10: $10^3 \times 10^6 = 10^9$. Combining these gives $1.8 \times 10^9$ joules. Looking at the wrong answers: Choice A ($1.8 \times 10^8$) results from miscalculating $500 \times 3.6$ as 180 instead of 1800, dropping a factor of 10. Choice C ($1.8 \times 10^{10}$) comes from incorrectly adding the exponents as $3 + 6 + 1 = 10$ instead of recognizing that 1800 should be written as $1.8 \times 10^3$. Choice D ($1.8 \times 10^{11}$) represents a more significant calculation error, possibly confusing the conversion factor or mishandling the scientific notation entirely. For unit conversion problems on the PSAT, always write out your dimensional analysis clearly and double-check your scientific notation arithmetic. Convert regular numbers to scientific notation first (like 500 = $5 \times 10^2$), then multiply coefficients and add exponents separately to avoid careless errors.
12. A swimmer completes $1{,}200$ meters in $15$ minutes. What is the swimmer's speed in kilometers per hour? Use $1000$ m $= 1$ km and $60$ min $= 1$ hr.
    Answer: 4.8 km/hr The question asks for the swimmer's speed in kilometers per hour, given 1,200 meters completed in 15 minutes, using 1,000 m = 1 km and 60 min = 1 hr. Set up the conversion by finding speed in m/min and applying factors 60 min/hr and 1 km/1,000 m. Speed is 1,200 m / 15 min = 80 m/min. Then, 80 m/min × (60 min/hr) × (1 km/1,000 m) = (80 × 60 / 1,000) km/hr = 4.8 km/hr, with minutes and meters canceling to leave km/hr. A common error is forgetting to convert minutes to hours, leading to 80 m/min × (1 km/1,000 m) = 0.08 km/min, then perhaps multiplying incorrectly. Another mistake is using 100 min/hr instead of 60. Always use dimensional analysis in rate conversions to ensure all units cancel to the desired ones.
13. A car travels at $55$ miles per hour. What is this speed in feet per second? Use $1 \text{ mi} = 5280 \text{ ft}$ and $1 \text{ hr} = 3600 \text{ s}$.
    Answer: 80.7 ft/s The question asks for the speed of a car traveling at $55$ miles per hour converted to feet per second, using $1 \text{ mile} = 5280 \text{ feet}$ and $1 \text{ hour} = 3600 \text{ seconds}$. Set up the conversion with factors $5280 \text{ feet per mile}$ for distance and $3600 \text{ seconds per hour}$ for time. Using dimensional analysis, start with $55 \text{ mi/hr}$ and multiply by ($5280 \text{ ft} / 1 \text{ mi}$) and ($1 \text{ hr} / 3600 \text{ s}$), ensuring miles cancel with miles and hours cancel with hours, leaving feet per second. Calculate $55 \times 5280 = 290,400$, then divide by 3600: $290,400 \div 3600 = 80.666\ldots \text{ ft/s}$, which rounds to 80.7 ft/s when considering significant figures or approximation. Common errors include forgetting to convert time units or inverting the time factor, leading to large numbers like 290,400 ft/s. Another mistake is incorrect simplification of fractions before calculating. A useful strategy is to perform dimensional analysis with all units visible to ensure proper cancellation and catch inversion errors early.
14. A bag of flour weighs $5\tfrac{1}{2}$ pounds. How many ounces is this? Use $1\text{ lb}=16\text{ oz}$.
    Answer: 88 oz We need to convert 5½ pounds to ounces. First express the mixed number as a decimal: 5½ = 5.5 pounds. Set up the conversion: 5.5 lb × (16 oz/1 lb) = 88 ounces. The pound units cancel, leaving ounces. A common error is multiplying only the whole number part (5 × 16 = 80 oz) and forgetting the fraction. When working with mixed numbers, convert to decimals or improper fractions first.
15. A sample of steel has a density of 7.8 grams per cubic centimeter. What is this density in kilograms per cubic meter?
    Answer: 7,800 kg/m³ When you encounter unit conversion problems involving density or other compound units, you need to convert each component separately and then combine them. To convert 7.8 grams per cubic centimeter to kilograms per cubic meter, you'll work with both the mass units (grams to kilograms) and volume units (cubic centimeters to cubic meters). First, convert grams to kilograms: Since 1 kilogram = 1,000 grams, you have $\frac{7.8 \text{ g}}{1000} = 0.0078 \text{ kg}$ Next, convert cubic centimeters to cubic meters: Since 1 meter = 100 centimeters, then 1 cubic meter = $(100)^3 = 1,000,000$ cubic centimeters. Now set up the conversion: $7.8 \frac{\text{g}}{\text{cm}^3} \times \frac{1 \text{ kg}}{1000 \text{ g}} \times \frac{1,000,000 \text{ cm}^3}{1 \text{ m}^3} = 7,800 \frac{\text{kg}}{\text{m}^3}$ The answer is C) 7,800 kg/m³. Looking at the wrong answers: A) 78 kg/m³ results from converting grams to kilograms but forgetting the volume conversion entirely. B) 780 kg/m³ comes from incorrectly using 100 instead of 1,000,000 for the volume conversion factor. D) 78,000 kg/m³ happens when you multiply by conversion factors in the wrong direction. Strategy tip: For compound units, always convert each component separately, then multiply all conversion factors together. Double-check that larger base units (like meters vs. centimeters) should give you larger final numbers for density.
16. A recipe needs $3.5$ quarts of broth. How many gallons is this? (Use $4\text{ quarts}=1\text{ gallon}$.)
    Answer: 0.875 gal We need to convert 3.5 quarts to gallons using the conversion factor 4 quarts = 1 gallon. Set up the conversion: 3.5 quarts × (1 gallon/4 quarts) = 3.5/4 gallons = 0.875 gallons. The quarts units cancel, leaving gallons. A common error is multiplying by 4 instead of dividing, which would give 14 gallons. When converting to a larger unit (gallons are larger than quarts), your numerical answer should be smaller than what you started with.
17. A cyclist rides at a constant speed of $18$ miles per hour. Using $1\text{ mile}=5280\text{ ft}$ and $1\text{ hour}=3600\text{ s}$, what is the cyclist's speed in feet per second?
    Answer: 26.4 ft/s We need to convert the cyclist's speed from miles per hour to feet per second. Set up the conversion using the given factors: 18 miles/hour × (5280 ft/1 mile) × (1 hour/3600 s). When we multiply, the miles cancel and hours cancel: 18 × 5280 ft ÷ 3600 s = 95,040 ft ÷ 3600 s = 26.4 ft/s. A common error is dividing by 5280 or multiplying by 3600 instead of dividing, which would give incorrect results. For quick estimation, remember that 1 mph ≈ 1.47 ft/s, so 18 mph should be about 26-27 ft/s.
18. A recipe needs 3.5 quarts of soup, but the pot is marked in cups. Using $1\text{ qt}=2\text{ pt}$ and $1\text{ pt}=2\text{ cups}$, how many cups of soup are needed?
    Answer: 14 cups The question asks to convert 3.5 quarts of soup to cups, using 1 quart = 2 pints and 1 pint = 2 cups. Set up the conversion by multiplying 3.5 qt by 2 pt/qt and then by 2 cups/pt to reach the desired unit. In dimensional analysis: 3.5 qt × (2 pt / 1 qt) × (2 cups / 1 pt) shows quarts and pints canceling, leaving cups. Compute 3.5 × 2 = 7, then 7 × 2 = 14 cups, or directly 3.5 × 4 = 14 cups, with units tracking throughout. A frequent error is using only one conversion factor, like just quarts to pints, resulting in 7 cups as in A. Another mistake might be reversing the factors, leading to fractions like D. As a strategy, list all steps with units to catch if intermediate conversions are missed.
19. A container holds 2.5 liters of juice. How many cubic centimeters of juice does the container hold? (1 liter = 1,000 cubic centimeters)
    Answer: 2,500 cm³ Unit conversion problems like this test your ability to set up proportions and apply conversion factors correctly. The key is identifying what you're converting from and to, then using the given relationship systematically. You have 2.5 liters and need to find the equivalent in cubic centimeters. Since 1 liter = 1,000 cubic centimeters, you can set up the conversion: $2.5 \text{ liters} \times \frac{1,000 \text{ cm}^3}{1 \text{ liter}} = 2,500 \text{ cm}^3$. The liters cancel out, leaving you with cubic centimeters. This confirms answer choice B is correct. Looking at the wrong answers reveals common conversion mistakes. Choice A (250 cm³) comes from dividing instead of multiplying—this happens when students mistakenly think they need to make the number smaller when converting to a "smaller" unit. Choice C (25,000 cm³) results from multiplying by 10,000 instead of 1,000, likely from misremembering the conversion factor or adding an extra zero. Choice D (250,000 cm³) comes from multiplying by 100,000, which might happen if you confuse this conversion with others or make multiple calculation errors. For unit conversion success on the PSAT, always write out your conversion factor as a fraction, ensuring the units you're converting from appear in both numerator and denominator so they cancel properly. This prevents the common mistake of dividing when you should multiply, and helps you double-check that your final units match what the question asks for.
20. A recipe uses $2.5$ gallons of soup. How many cups is this? Use $1\text{ gallon}=4\text{ quarts}$ and $1\text{ quart}=4\text{ cups}$.
    Answer: 40 cups We need to convert 2.5 gallons to cups. Setting up the conversion with two steps: 2.5 gallons × (4 quarts/1 gallon) × (4 cups/1 quart). First convert to quarts: 2.5 × 4 = 10 quarts. Then convert to cups: 10 × 4 = 40 cups. Notice how the units cancel: gallons → quarts → cups. A common mistake is using only one conversion factor and getting 10 cups. Remember to chain conversions when going through intermediate units.

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