Use dimensional analysis to set up conversion problems - Health Education Systems Admission Assessment HESI

1. A physician orders heparin to be infused at 1,100 units/hour. The pharmacy provides a bag of 500 mL solution containing 25,000 units of heparin. Which dimensional analysis setup correctly calculates the required IV pump rate in mL/hour?
   Answer: $\frac{1100 \text{ units}}{1 \text{ hr}} \times \frac{500 \text{ mL}}{25000 \text{ units}}$ When you encounter IV drip calculations, you're working with dimensional analysis to convert between different units while ensuring unwanted units cancel out. The key is setting up your conversion factors so that units cancel properly and you end up with the desired final unit. You need to convert from the ordered rate (1,100 units/hour) to the pump setting (mL/hour). Start with what you know: the ordered dose of 1,100 units per hour. Then you need a conversion factor that relates the concentration of your IV bag - you have 25,000 units dissolved in 500 mL. Answer B correctly sets up this conversion: $\frac{1100 \text{ units}}{1 \text{ hr}} \times \frac{500 \text{ mL}}{25000 \text{ units}}$. Notice how "units" appears in both the numerator of the first fraction and denominator of the second fraction, so they cancel out, leaving you with mL/hr. Answer A uses the concentration ratio upside down ($\frac{25000 \text{ units}}{500 \text{ mL}}$), which would give you units²/(hr × mL) - the wrong final unit entirely. Answer C flips the dose rate to $\frac{1 \text{ hr}}{1100 \text{ units}}$, resulting in hr²/units after cancellation - completely wrong units. Answer D combines both errors from A and C, again producing incorrect units that don't make sense for a pump rate. Study tip: Always check your dimensional analysis by writing out the units and confirming they cancel to give you the unit you want. If your setup doesn't produce mL/hr for pump rates, rearrange your conversion factors.
2. A patient's temperature increases by 2.5° Celsius. A nurse wants to document this change in degrees Fahrenheit. Which setup correctly calculates the magnitude of the temperature change in Fahrenheit? (The conversion factor for temperature change is 9°F / 5°C).
   Answer: $2.5 \times \frac{9}{5}$ When you encounter temperature conversion problems in nursing, it's crucial to distinguish between converting absolute temperatures versus temperature changes. For absolute temperature conversions, you use the full formula with the +32 offset, but for temperature changes (increases or decreases), you only need the conversion factor. The correct approach is answer D: $2.5 \times \frac{9}{5}$. Since you're converting a temperature change of 2.5°C to Fahrenheit, you simply multiply by the conversion factor 9°F/5°C. This gives you $2.5 \times \frac{9}{5} = 4.5°F$. The +32 adjustment is not needed because you're measuring the magnitude of change, not converting between absolute temperature scales. Answer A incorrectly adds 32 to the calculation: $(2.5 \times \frac{9}{5}) + 32$. This would give 36.5°F, which represents an absolute temperature, not a temperature change. Answer B makes the same error but applies the operations in the wrong order: $(2.5 + 32) \times \frac{9}{5}$. This completely distorts the conversion by adding 32 to the Celsius change first. Answer C uses the wrong conversion factor: $2.5 \times \frac{5}{9}$. This factor (5°C/9°F) would convert Fahrenheit changes to Celsius, not the reverse. Remember: when converting temperature changes, ignore the +32 offset entirely. Use only the 9/5 factor for Celsius to Fahrenheit changes, or 5/9 for Fahrenheit to Celsius changes. The +32 is only for converting absolute temperatures between scales.
3. A patient's urine output is measured as 1200 mL over a 24-hour period. The healthcare provider wants to know the output rate in drops per second, given that 1 mL equals approximately 20 drops. Which dimensional analysis setup correctly determines this rate?
   Answer: $1200 \text{ mL} \times \frac{1 \text{ day}}{24 \text{ hr}} \times \frac{1 \text{ hr}}{3600 \text{ sec}} \times \frac{20 \text{ drops}}{1 \text{ mL}}$ When you encounter dimensional analysis problems in healthcare calculations, you're converting units by multiplying by conversion factors arranged so unwanted units cancel out, leaving only your desired units. Let's work through this systematically. You need to convert 1200 mL over 24 hours to drops per second. Start with what you have (1200 mL) and set up conversion factors to reach your target units (drops/second). The correct setup is option D: $1200 \text{ mL} \times \frac{1 \text{ day}}{24 \text{ hr}} \times \frac{1 \text{ hr}}{3600 \text{ sec}} \times \frac{20 \text{ drops}}{1 \text{ mL}}$ This works because: the "mL" cancels with the last fraction, "day" cancels between the first and second fractions, "hr" cancels between the second and third fractions, leaving only "drops/sec" as desired. Option A incorrectly inverts the drops-to-mL conversion ($\frac{1 \text{ mL}}{20 \text{ drops}}$), which would give you the wrong units and a much smaller number. Option B uses $\frac{24 \text{ hr}}{1 \text{ day}}$ and $\frac{3600 \text{ sec}}{1 \text{ hr}}$, which multiplies by time instead of dividing by it. This would give you total drops over many hours rather than drops per second. Option C has the same time conversion error as B, multiplying rather than dividing by the time factors. Study tip: In dimensional analysis, always check that unwanted units cancel out completely. Write out the units throughout your calculation to catch errors before computing the final number.
4. A nurse must calculate the flow rate for an IV infusion. The order is for 1000 mL to be administered over 8 hours using tubing with a drop factor of 15 gtts/mL. Which dimensional analysis expression will yield the correct drip rate in drops per minute?
   Answer: $1000 \text{ mL} \times \frac{1 \text{ hr}}{8 \text{ hr}} \times \frac{15 \text{ gtts}}{1 \text{ mL}} \times \frac{1 \text{ hr}}{60 \text{ min}}$ Choice A correctly sets up the calculation: 1000 mL × (1 hr/8 hr) gives mL per hour, × (15 gtts/1 mL) converts to gtts per hour, × (1 hr/60 min) converts to gtts per minute. Choice B inverts the time conversion (8 hr/1 hr) which would give an incorrect rate. Choice C has the minute conversion inverted (60 min/1 hr) which would give gtts per hour squared. Choice D has multiple errors: inverted time factor and inverted drop factor.
5. A patient is receiving a continuous medication infusion at 15 mcg/kg/min. The patient weighs 176 pounds and the medication concentration is 400 mcg/mL. Which dimensional analysis setup correctly calculates the infusion rate in mL per hour?
   Answer: $176 \text{ lb} \times \frac{1 \text{ kg}}{2.2 \text{ lb}} \times \frac{15 \text{ mcg}}{1 \text{ kg} \cdot \text{min}} \times \frac{60 \text{ min}}{1 \text{ hr}} \times \frac{1 \text{ mL}}{400 \text{ mcg}}$ Choice A correctly sets up the conversion: convert pounds to kg, multiply by dose rate to get mcg/min, multiply by 60 min/hr to get mcg/hr, then convert to mL/hr using the concentration. Choice B has the kg conversion inverted and uses 1 hr/60 min instead of 60 min/hr. Choice C has the time conversion inverted (1 hr/60 min) which would give mcg per minute squared. Choice D has the concentration factor inverted (400 mcg/1 mL instead of 1 mL/400 mcg).
6. A patient is receiving a dopamine infusion at 7 mcg/kg/min. The patient weighs 165 lbs. The solution concentration is 400 mg in 250 mL D5W. Which expression correctly sets up the calculation for the infusion pump rate in mL/hr? (1 kg ≈ 2.2 lbs; 1 mg = 1000 mcg)
   Answer: $\frac{7 \text{ mcg}}{1 \text{ kg} \cdot \text{min}} \times \frac{165 \text{ lbs}}{1} \times \frac{1 \text{ kg}}{2.2 \text{ lbs}} \times \frac{1 \text{ mg}}{1000 \text{ mcg}} \times \frac{250 \text{ mL}}{400 \text{ mg}} \times \frac{60 \text{ min}}{1 \text{ hr}}$ IV medication dosage calculations require systematic dimensional analysis to convert between units while maintaining proper direction of conversion factors. When setting up these problems, you must carefully track units to ensure they cancel correctly, leading you from the starting dose to the final pump rate. The correct setup in option A follows logical steps: Start with the prescribed dose (7 mcg/kg/min), multiply by the patient's weight (165 lbs), convert pounds to kilograms using $\frac{1 \text{ kg}}{2.2 \text{ lbs}}$, convert micrograms to milligrams using $\frac{1 \text{ mg}}{1000 \text{ mcg}}$, convert milligrams to volume using the solution concentration $\frac{250 \text{ mL}}{400 \text{ mg}}$, and finally convert minutes to hours using $\frac{60 \text{ min}}{1 \text{ hr}}$. Option B contains two critical errors: it uses the inverted conversion $\frac{1000 \text{ mcg}}{1 \text{ mg}}$ (which converts mg to mcg instead of mcg to mg) and $\frac{1 \text{ hr}}{60 \text{ min}}$ (converting hours to minutes instead of minutes to hours). Option C has multiple errors: $\frac{2.2 \text{ lbs}}{1 \text{ kg}}$ converts kg to lbs (backward), and $\frac{400 \text{ mg}}{250 \text{ mL}}$ converts mL to mg instead of mg to mL. Option D incorrectly places the patient's weight in the denominator as $\frac{1}{165 \text{ lbs}}$, which would divide by the weight instead of multiplying. Study tip: Always write out the units in dimensional analysis problems and verify they cancel to give your desired final units. Each conversion factor should move you one step closer to mL/hr.
7. A pediatric patient requires an antibiotic at a dose of 30 mg/kg/day. The child weighs 33 lbs and the oral suspension is available at a concentration of 250 mg/5 mL. The physician wants to know the total volume in mL the child will receive over a 24-hour period. Which setup is correct? (1 kg = 2.2 lbs)
   Answer: $\frac{33 \text{ lbs}}{1} \times \frac{1 \text{ kg}}{2.2 \text{ lbs}} \times \frac{30 \text{ mg}}{1 \text{ kg}} \times \frac{5 \text{ mL}}{250 \text{ mg}}$ Pediatric dosage calculations require systematic dimensional analysis to convert between units and ensure accurate medication administration. When you encounter these problems, identify what you're given, what you need to find, and set up conversion factors so units cancel properly. You need to find the total volume in mL for a 24-hour period. Starting with the child's weight of 33 lbs, you must: convert pounds to kilograms, calculate the total daily dose in mg, then convert mg to mL using the suspension concentration. The correct setup in option B follows this logical sequence: $\frac{33 \text{ lbs}}{1} \times \frac{1 \text{ kg}}{2.2 \text{ lbs}} \times \frac{30 \text{ mg}}{1 \text{ kg}} \times \frac{5 \text{ mL}}{250 \text{ mg}}$. Notice how units cancel: lbs cancels with lbs, kg cancels with kg, and mg cancels with mg, leaving only mL as your final answer. Option A incorrectly uses $\frac{2.2 \text{ lbs}}{1 \text{ kg}}$ instead of $\frac{1 \text{ kg}}{2.2 \text{ lbs}}$, which would multiply the weight rather than convert it properly to kilograms. Option C has $\frac{1 \text{ kg}}{30 \text{ mg}}$ instead of $\frac{30 \text{ mg}}{1 \text{ kg}}$, placing the dose conversion upside down. Option D uses $\frac{250 \text{ mg}}{5 \text{ mL}}$ instead of $\frac{5 \text{ mL}}{250 \text{ mg}}$, inverting the final concentration conversion. Remember: in dimensional analysis, always check that your conversion factors are oriented so unwanted units cancel out, leaving only the units you want in your final answer.
8. A patient's height is measured as 5 feet, 8 inches. To calculate BMI, the height must be in meters. Which expression correctly sets up the conversion of the total height in inches to meters? (Use 1 inch = 2.54 cm and 1 m = 100 cm)
   Answer: $(5 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} + 8 \text{ in}) \times \frac{2.54 \text{ cm}}{1 \text{ in}} \times \frac{1 \text{ m}}{100 \text{ cm}}$ Unit conversion problems in healthcare require systematic dimensional analysis to ensure accurate calculations for clinical assessments like BMI. The key is setting up conversion factors so that unwanted units cancel out, leaving only the desired unit. To convert 5 feet, 8 inches to meters, you need a three-step process. First, convert the mixed units (feet and inches) to total inches: $5 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} + 8 \text{ in} = 68 \text{ inches}$. Then convert inches to centimeters using the given conversion factor: $\frac{2.54 \text{ cm}}{1 \text{ in}}$. Finally, convert centimeters to meters: $\frac{1 \text{ m}}{100 \text{ cm}}$. Choice D correctly combines all these steps: $(5 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} + 8 \text{ in}) \times \frac{2.54 \text{ cm}}{1 \text{ in}} \times \frac{1 \text{ m}}{100 \text{ cm}}$. Notice how the units cancel properly: inches cancel between the first and second terms, centimeters cancel between the second and third terms, leaving meters. Choice A skips the centimeter step and uses an incorrect conversion factor. Choice B incorrectly adds feet and inches before converting feet to inches, which is mathematically invalid since you can't add different units directly. Choice C has the conversion factor for the final step inverted ($\frac{100 \text{ cm}}{1 \text{ m}}$), which would multiply by 100 instead of dividing, giving an answer 10,000 times too large. For HESI dimensional analysis problems, always write out each conversion step separately and verify that units cancel properly before calculating the final answer.
9. To prepare a cleaning solution, a nurse must dilute a concentrate. The instructions state to use 3 fluid ounces (fl oz) of concentrate for every 1 gallon (gal) of water. Which expression correctly sets up the calculation for how many milliliters (mL) of concentrate are needed to prepare a 5-gallon solution? (Use 1 fl oz ≈ 29.57 mL)
   Answer: $\frac{5 \text{ gal}}{1} \times \frac{3 \text{ fl oz}}{1 \text{ gal}} \times \frac{29.57 \text{ mL}}{1 \text{ fl oz}}$ When you encounter dosage calculation problems involving unit conversions, you need to set up a dimensional analysis chain that systematically cancels unwanted units while preserving the desired final unit. The correct approach is option B: $\frac{5 \text{ gal}}{1} \times \frac{3 \text{ fl oz}}{1 \text{ gal}} \times \frac{29.57 \text{ mL}}{1 \text{ fl oz}}$. This expression starts with 5 gallons of total solution, then applies the concentration ratio (3 fl oz concentrate per 1 gallon), and finally converts fluid ounces to milliliters. Notice how the units cancel perfectly: gal cancels with gal, and fl oz cancels with fl oz, leaving only mL. Option A inverts the concentration ratio incorrectly, showing $\frac{1 \text{ gal}}{3 \text{ fl oz}}$ instead of $\frac{3 \text{ fl oz}}{1 \text{ gal}}$. This would give you gallons per fluid ounce rather than the needed fluid ounces of concentrate per gallon of solution. Option C uses the correct concentration ratio but inverts the conversion factor, showing $\frac{1 \text{ fl oz}}{29.57 \text{ mL}}$ instead of $\frac{29.57 \text{ mL}}{1 \text{ fl oz}}$. This would convert milliliters to fluid ounces rather than fluid ounces to milliliters. Option D places 5 gallons in the denominator ($\frac{1}{5 \text{ gal}}$), which would divide by the volume instead of multiplying by it, giving an incorrect result. For HESI dosage calculations, always write out your dimensional analysis completely and check that unwanted units cancel. Set up each conversion factor so the unit you want to eliminate appears in both a numerator and denominator.
10. A medication order calls for 0.75 mg/kg/day divided into three equal doses. For a 132-pound patient, if the medication comes as 25 mg tablets that can be split in half, which setup correctly determines how many tablet portions are needed per dose?
    Answer: $132 \text{ lb} \times \frac{1 \text{ kg}}{2.2 \text{ lb}} \times \frac{0.75 \text{ mg}}{1 \text{ kg} \cdot \text{day}} \times \frac{1 \text{ day}}{3 \text{ doses}} \times \frac{1 \text{ tablet}}{25 \text{ mg}}$ Choice A correctly calculates tablets per dose: convert 132 lb to kg, multiply by 0.75 mg/kg/day for total daily dose, divide by 3 doses per day to get mg per dose, then convert to tablets using 1 tablet/25 mg. Choice B has the kg conversion inverted and multiplies by 3 instead of dividing. Choice C multiplies by 3 doses instead of dividing, giving total daily tablets rather than per dose. Choice D has the tablet conversion factor inverted.
11. A protocol requires a saline flush of 2.5 mL to be administered over 30 seconds. Which expression correctly sets up the calculation to find the equivalent flow rate in liters per hour (L/hr)?
    Answer: $\frac{2.5 \text{ mL}}{30 \text{ sec}} \times \frac{1 \text{ L}}{1000 \text{ mL}} \times \frac{3600 \text{ sec}}{1 \text{ hr}}$ Unit conversion problems in healthcare require careful attention to setting up dimensional analysis correctly. When converting flow rates, you need to systematically convert each unit while maintaining the proper mathematical relationships. Starting with the given rate of 2.5 mL over 30 seconds, you first express this as $\frac{2.5 \text{ mL}}{30 \text{ sec}}$. To convert to L/hr, you need two additional conversion factors: milliliters to liters, and seconds to hours. Since 1 L = 1000 mL, you multiply by $\frac{1 \text{ L}}{1000 \text{ mL}}$ to convert the numerator. Since 1 hour = 3600 seconds, you multiply by $\frac{3600 \text{ sec}}{1 \text{ hr}}$ to convert the denominator. This gives you option A, which correctly sets up the calculation. Option B uses the wrong conversion factors - it has $\frac{1000 \text{ mL}}{1 \text{ L}}$ (which would convert liters to milliliters, not the reverse) and $\frac{1 \text{ hr}}{3600 \text{ sec}}$ (which would convert hours to seconds). Option C flips the initial rate to $\frac{30 \text{ sec}}{2.5 \text{ mL}}$, which would give you the reciprocal of flow rate. Option D stops the time conversion at minutes rather than completing the conversion to hours, using only $\frac{60 \text{ sec}}{1 \text{ min}}$. Remember: in dimensional analysis, units must cancel properly. Write out all units in your conversion factors and verify that unwanted units cancel, leaving only your desired final units.
12. A patient consumes a meal containing 40 grams of carbohydrates. The nurse needs to calculate the energy provided in kilojoules (kJ). Which setup is correct? (Use 1 gram of carbohydrate ≈ 4 kcal; 1 kcal ≈ 4.184 kJ)
    Answer: $\frac{40 \text{ g}}{1} \times \frac{4 \text{ kcal}}{1 \text{ g}} \times \frac{4.184 \text{ kJ}}{1 \text{ kcal}}$ When you encounter energy conversion problems in healthcare, you need to carefully set up dimensional analysis to convert step-by-step from your starting unit to your target unit. The key is ensuring each conversion factor is positioned so unwanted units cancel out. You're converting 40 grams of carbohydrates to kilojoules using two given conversion factors: 1 gram carbohydrate = 4 kcal, and 1 kcal = 4.184 kJ. Start with 40 grams, then multiply by conversion factors that will eliminate grams and kcal, leaving only kJ. The correct setup is $\frac{40 \text{ g}}{1} \times \frac{4 \text{ kcal}}{1 \text{ g}} \times \frac{4.184 \text{ kJ}}{1 \text{ kcal}}$. This works because grams cancel (40 g × g⁻¹), then kcal cancel (kcal × kcal⁻¹), leaving only kJ. The calculation gives: 40 × 4 × 4.184 = 669.44 kJ. Option A incorrectly places grams in the denominator of the first conversion (1 g/4 kcal), which would give you kcal/g instead of eliminating grams. Option B has the same gram problem, plus it inverts the final conversion factor (1 kcal/4.184 kJ), which would cancel out kcal and leave you with an inverted kJ unit. Option D correctly handles the final conversion but makes the same initial error as option A with the gram placement. Remember: in dimensional analysis, the unit you want to eliminate must appear in both numerator and denominator positions so they cancel. Always trace through your setup to verify that unwanted units disappear and only your target unit remains.
13. A patient requires 1800 calories per day, and 60% of these calories should come from a tube feeding formula that provides 1.5 cal/mL. The feeding will be given continuously over 18 hours per day. Which setup correctly calculates the hourly infusion rate in mL per hour?
    Answer: $1800 \text{ cal/day} \times \frac{60 \text{ cal from formula}}{100 \text{ total cal}} \times \frac{1 \text{ day}}{18 \text{ hr}} \times \frac{1 \text{ mL}}{1.5 \text{ cal}}$ When you encounter tube feeding calculations, you need to systematically convert from total daily calories to an hourly infusion rate by accounting for the percentage from formula, the formula concentration, and the infusion duration. Let's work through this step-by-step. You start with 1800 calories per day total. Since 60% comes from tube feeding, you need: $1800 \times \frac{60}{100} = 1080$ calories from formula daily. Next, convert calories to volume using the formula concentration of 1.5 cal/mL: $1080 \text{ cal} \times \frac{1 \text{ mL}}{1.5 \text{ cal}} = 720 \text{ mL}$ total volume needed. Finally, since this 720 mL is given over 18 hours, the hourly rate is: $\frac{720 \text{ mL}}{18 \text{ hr}} = 40 \text{ mL/hr}$. Choice C correctly sets up this calculation in one equation: $1800 \times \frac{60}{100} \times \frac{1}{18} \times \frac{1}{1.5}$ Choice A uses 40% instead of 60%, which would incorrectly calculate calories from other sources rather than the tube feeding. Choice B multiplies by $\frac{24 \text{ hr}}{1 \text{ day}}$ instead of dividing by 18 hours, which increases the rate rather than distributing the daily volume over the actual infusion time. Choice D multiplies by $\frac{18 \text{ hr}}{1 \text{ day}}$ instead of dividing, which would give you total daily volume times 18 rather than the hourly rate. Remember: tube feeding calculations always follow the pattern of total calories → formula calories → volume → hourly rate. Double-check that you're dividing by infusion hours, not multiplying.
14. A patient needs to receive 2.5 mg of medication per kilogram of body weight. The patient weighs 154 pounds and the medication is available in a concentration of 10 mg/mL. Which dimensional analysis setup correctly determines the volume of medication needed?
    Answer: $154 \text{ lb} \times \frac{1 \text{ kg}}{2.2 \text{ lb}} \times \frac{2.5 \text{ mg}}{1 \text{ kg}} \times \frac{1 \text{ mL}}{10 \text{ mg}}$ Choice A correctly sets up the dimensional analysis. Starting with 154 lb, multiply by (1 kg/2.2 lb) to convert to kg, then by (2.5 mg/1 kg) to get total mg needed, then by (1 mL/10 mg) to convert to volume. Choice B uses the wrong conversion factor for pounds to kilograms (inverted). Choice C has the final concentration factor inverted. Choice D has both the weight conversion inverted and uses lb/kg instead of kg/lb.
15. A patient's lab result for creatinine is 106 micromol/L (µmol/L). A different hospital system records this value in mg/dL. Which setup correctly converts the value? (Molecular weight of creatinine ≈ 113 g/mol; 1 mol = 1,000,000 µmol; 1 g = 1000 mg; 1 L = 10 dL)
    Answer: $\frac{106 \text{ µmol}}{1 \text{ L}} \times \frac{1 \text{ mol}}{10^6 \text{ µmol}} \times \frac{113 \text{ g}}{1 \text{ mol}} \times \frac{1000 \text{ mg}}{1 \text{ g}} \times \frac{1 \text{ L}}{10 \text{ dL}}$ Unit conversion problems in healthcare require systematic dimensional analysis to ensure patient safety. When converting between different measurement systems, you must carefully track each conversion factor to cancel units properly and arrive at the correct final units. To convert 106 µmol/L to mg/dL, you need to transform both the numerator (µmol to mg) and denominator (L to dL). Starting with $\frac{106 \text{ µmol}}{1 \text{ L}}$, the correct sequence is: convert µmol to mol, then mol to grams using molecular weight, then grams to milligrams, and finally liters to deciliters. Choice A correctly executes this conversion: $\frac{1 \text{ mol}}{10^6 \text{ µmol}}$ converts micromoles to moles, $\frac{113 \text{ g}}{1 \text{ mol}}$ uses the molecular weight properly, $\frac{1000 \text{ mg}}{1 \text{ g}}$ converts to milligrams, and $\frac{1 \text{ L}}{10 \text{ dL}}$ converts the volume units correctly. Choice B inverts the µmol to mol conversion ($\frac{10^6 \text{ µmol}}{1 \text{ mol}}$) and the mg conversion ($\frac{1 \text{ g}}{1000 \text{ mg}}$), which would multiply instead of converting properly. Choice C incorrectly inverts the molecular weight factor ($\frac{1 \text{ mol}}{113 \text{ g}}$), treating it as if molecular weight converts grams to moles rather than moles to grams. Choice D inverts the milligram conversion, which would convert mg to grams instead of grams to mg. Remember: in dimensional analysis, write conversion factors so unwanted units cancel out. Always double-check that your setup will yield the target units before calculating.
16. A medication has a concentration of 250 mg per 5 mL and is to be administered at 12 mg/kg every 8 hours. For a 45 kg patient, which dimensional analysis expression correctly determines the volume needed per dose?
    Answer: $45 \text{ kg} \times \frac{12 \text{ mg}}{1 \text{ kg} \cdot \text{dose}} \times \frac{5 \text{ mL}}{250 \text{ mg}}$ Dimensional analysis questions test your ability to set up conversion factors so that unwanted units cancel out, leaving only the desired unit. When you see dosage calculations, focus on what the question is asking for and work systematically through the conversions. The question asks for the volume per dose, so you need to convert from the patient's weight to the final volume in mL. Start with 45 kg, then convert to mg needed per dose using the dosing rate of 12 mg/kg/dose. Finally, convert from mg to mL using the concentration. Choice C correctly sets up this pathway: $45 \text{ kg} \times \frac{12 \text{ mg}}{1 \text{ kg} \cdot \text{dose}} \times \frac{5 \text{ mL}}{250 \text{ mg}}$. The units cancel properly (kg cancels with kg, mg cancels with mg), leaving mL/dose as the final unit. Choice A uses the concentration fraction upside down ($\frac{250 \text{ mg}}{5 \text{ mL}}$), which would give you an answer in mg²/(kg·mL·dose) instead of mL/dose. Choice B unnecessarily includes the 8-hour frequency, but since the question asks for volume per dose (not per hour), this extra conversion factor makes the setup incorrect. Choice D also incorrectly includes the 8-hour timing and puts it in the wrong position, which would multiply rather than divide by the frequency. For HESI dosage calculations, always identify your starting point, your desired endpoint, and set up conversion factors so units cancel step by step. Ignore information that doesn't relate to what the question is specifically asking for.
17. A medical device's pressure gauge reads 120 kilopascals (kPa). A manual specifies the operating pressure in pounds per square inch (psi). Which dimensional analysis setup correctly converts the pressure from kPa to psi? (Use 1 kPa = 0.145 psi)
    Answer: $\frac{120 \text{ kPa}}{1} \times \frac{0.145 \text{ psi}}{1 \text{ kPa}}$ Dimensional analysis questions test your ability to set up unit conversions correctly by canceling unwanted units and keeping desired ones. When converting between units, you must arrange conversion factors so that units cancel algebraically, leaving only the target unit. To convert 120 kPa to psi, you need to eliminate kPa and retain psi. Starting with your given value (120 kPa), you multiply by a conversion factor where kPa appears in the denominator (to cancel with the kPa in your starting value) and psi appears in the numerator (your desired final unit). Since 1 kPa = 0.145 psi, the correct setup is $\frac{120 \text{ kPa}}{1} \times \frac{0.145 \text{ psi}}{1 \text{ kPa}}$. The kPa units cancel, leaving 120 × 0.145 = 17.4 psi. Choice A sets up the conversion factor backwards ($\frac{1 \text{ psi}}{0.145 \text{ kPa}}$), which would give an incorrect numerical relationship and wrong final units. Choice C incorrectly places 120 kPa in the denominator of the first fraction, turning your starting value into a reciprocal rather than preserving it as the quantity being converted. Choice D simply adds the values together, which is meaningless since you cannot add different units—this completely ignores the conversion process. For dimensional analysis success, always write out units in your setup and verify they cancel to give your target unit. The numbers should follow the unit relationships, not the other way around.
18. A medication's half-life is 8 hours. A patient is given a dose of 500 mg. Which setup correctly calculates how many days it will take for the amount of medication in the body to be reduced to 62.5 mg? (Note: 62.5 mg is 1/8th of 500 mg, which represents 3 half-lives).
    Answer: $\frac{3 \text{ half-lives}}{1} \times \frac{8 \text{ hours}}{1 \text{ half-life}} \times \frac{1 \text{ day}}{24 \text{ hours}}$ When you encounter medication half-life problems, you're working with pharmacokinetics—how drugs are processed and eliminated from the body. The key is setting up dimensional analysis to convert between different units while tracking the biological process. Half-life represents the time needed for a drug concentration to decrease by 50%. To go from 500 mg to 62.5 mg, you need to determine how many half-lives this represents: 500 → 250 → 125 → 62.5 mg. That's 3 half-lives, as the question confirms. The correct setup is C: $\frac{3 \text{ half-lives}}{1} \times \frac{8 \text{ hours}}{1 \text{ half-life}} \times \frac{1 \text{ day}}{24 \text{ hours}}$. This multiplies the number of half-lives needed (3) by the duration of each half-life (8 hours), then converts to days. The units cancel properly: half-lives cancel, hours cancel, leaving days. Choice A incorrectly places "8 hours" in the denominator of the second fraction, which would divide by 8 instead of multiply, giving an impossibly small result. Choice B starts with the mg ratio (500/62.5 = 8), but this represents the fold-reduction, not the number of half-lives—it's the wrong starting point. Choice D begins with 62.5 mg, which has no logical connection to calculating time duration. For HESI pharmacology questions, always identify what you're solving for, then work systematically through dimensional analysis. Make sure your units cancel correctly—if you're calculating time, your final units should be time-related, not concentration-related.
19. A powdered medication must be reconstituted. The vial contains 1 gram of powder. Instructions state to add 9.6 mL of sterile water to yield a solution with a concentration of 100 mg/mL. A dose of 250 mg is ordered. Which expression correctly sets up the calculation for the volume to administer in mL?
    Answer: $\frac{250 \text{ mg}}{1} \times \frac{1 \text{ mL}}{100 \text{ mg}}$ When you encounter medication dosage calculations involving reconstituted powders, focus on the final concentration after mixing, not the original powder amount or the diluent volume used during reconstitution. The key information here is that after reconstitution, you have a solution with a concentration of 100 mg/mL, and you need to deliver 250 mg. This is a straightforward dose-to-volume conversion using dimensional analysis. Answer A correctly sets up the calculation: $\frac{250 \text{ mg}}{1} \times \frac{1 \text{ mL}}{100 \text{ mg}}$. The milligrams cancel out, leaving you with the volume in mL. This gives you 2.5 mL, which is the correct volume to administer. Answer B has the concentration fraction inverted: $\frac{100 \text{ mg}}{1 \text{ mL}}$ instead of $\frac{1 \text{ mL}}{100 \text{ mg}}$. This setup would give you mg²/mL instead of mL, making it dimensionally incorrect. Answer C unnecessarily incorporates the original powder mass (1 g) and reconstitution volume (9.6 mL), which are irrelevant once you know the final concentration. This complex setup doesn't lead to the correct volume calculation. Answer D attempts to use the total reconstitution volume (9.6 mL) with the total drug amount (1000 mg), but this approach ignores that you only need a portion of the total solution to deliver the ordered dose. Study tip: For reconstituted medications, always use the final concentration (mg/mL) given in the problem. Ignore the reconstitution details once you have the final concentration—they're distractors designed to complicate a simple dose calculation.
20. A patient is receiving an insulin infusion at 8 units per hour. The insulin concentration is 100 units in 250 mL of normal saline. After 6.5 hours, the nurse needs to determine how much fluid volume the patient has received from this infusion. Which dimensional analysis expression correctly calculates this volume?
    Answer: $8 \frac{\text{units}}{\text{hr}} \times 6.5 \text{ hr} \times \frac{250 \text{ mL}}{100 \text{ units}}$ Dimensional analysis is a powerful tool for medication calculations that ensures you get the right units by systematically canceling them out. When setting up these problems, you must arrange conversion factors so that unwanted units cancel and you're left with the desired unit. To find the fluid volume received, you need to work through three steps: determine total units given, then convert units to volume. Start with the rate (8 units/hr) and multiply by time (6.5 hr) to get total units administered. Then convert those units to milliliters using the concentration ratio. Option B is correct: $8 \frac{\text{units}}{\text{hr}} \times 6.5 \text{ hr} \times \frac{250 \text{ mL}}{100 \text{ units}}$. The hours cancel out, leaving 52 units total. Then units cancel out using the proper concentration ratio, giving you 130 mL. Option A uses the concentration ratio upside down ($\frac{100 \text{ units}}{250 \text{ mL}}$), which would give you units²/mL instead of mL. The units don't cancel properly. Option C incorrectly puts 6.5 hours in the denominator ($\frac{1 \text{ hr}}{6.5 \text{ hr}}$), which would drastically reduce the time factor instead of multiplying by the total time. Option D unnecessarily adds a time conversion factor that isn't needed and would give incorrect units. Remember: in dimensional analysis, set up your fractions so units cancel diagonally. Always check that your final units match what the question asks for—if they don't, you've made a setup error.

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