Educational arithmetic only. All quantities below are invented CARE teaching examples, not patient orders, safe-dose ranges or administration instructions. Correct arithmetic does not establish clinical appropriateness. Use actual orders, product labeling, device limits and organizational policy in practice.
Make the requested unit your destination
Dimensional analysis arranges multiplication factors so unwanted units cancel and the requested unit remains. It helps make a setup inspectable: another person can follow what each number represents. It does not replace understanding the problem, checking its assumptions or checking the arithmetic.
- Write the requested unit before calculating: mL, tablets, mL/hour or minutes.
- List the given quantities with their complete units. Keep a concentration’s numerator and denominator together.
- Choose factors that place each unwanted unit opposite the same unit in the next factor.
- Cancel units first; multiply and divide second. Keep intermediate precision.
- Apply the exercise’s final-rounding instruction, label the result and check its size against the original relationship.
Method reference: Open RN Nursing Skills, chapter 5, sections 5.4–5.6. The examples and teaching sequence here are original CARE material.
1. Convert units before combining amounts
The metric prefixes describe powers of ten: 1 g = 1,000 mg and 1 mg = 1,000 micrograms. We write micrograms as mcg in these medication-context exercises. A gram-to-milligram conversion changes the number and unit, not the amount represented.
Convert 0.36 g into mg
0.36 g × (1,000 mg / 1 g) = 360 mg
The g cancels. A smaller unit requires a larger numerical count. Multiplying instead by 1 g / 1,000 mg would leave g²/mg, not mg: the unit check rejects that setup before a calculator is needed.
NIST metric prefixes supports these exact metric relationships. Do not treat common household or pound-to-kilogram approximations as exact metric definitions.
2. Orient a fictional concentration toward mL
A fictional solution contains 90 mg in 3 mL
For this arithmetic exercise, find the volume containing 150 mg. Give the exact result in mL.
150 mg × (3 mL / 90 mg) = 450 / 90 mL = 5 mL
mg cancels, leaving mL. Independently check: 90 mg / 3 mL = 30 mg/mL; 5 mL × 30 mg/mL = 150 mg. This verifies the relationship, not whether anyone should receive that amount.
Wrong-way factor: 150 mg × 90 mg / 3 mL leaves mg²/mL. A plausible-looking calculator result cannot rescue an incorrect unit setup.
Read the whole concentration. “90 mg in 3 mL” is not “90 mg per mL.” If the label or problem is incomplete, do not supply a missing denominator from habit.
Move into existing liquid-volume practice after you can explain both the cancellation and the inverse check.
3. Include the conversion that connects time units
A fictional 420 mL volume over 3.5 hours
Find the arithmetic rate in mL/hour, then in mL/minute. No rounding is needed.
(420 mL / 3.5 hours) = 120 mL/hour
120 mL/hour × (1 hour / 60 minutes) = 2 mL/minute
Check the total duration: 3.5 × 60 = 210 minutes; 2 mL/minute × 210 minutes = 420 mL. Do not multiply by 60 when converting this hourly rate into a per-minute rate: each minute represents a smaller share of the volume.
For a drops-per-minute exercise, a specified tubing factor in drops/mL is also needed. Never assume a drop factor. The same volume/time pair gives different numerical drop rates for different factors.
Related practice: mL/hour, drops/minute and duration. These are separate requested units, not interchangeable answers.
4. Preserve the interval: per day is not per dose
A fictional total of 720 mg per day divided into three equal portions
This example defines the portions as equal. Find mg per portion.
(720 mg/day) × (1 day/3 portions) = 240 mg/portion
Three portions × 240 mg/portion = 720 mg total. If 720 mg were specified per portion instead, the result would describe a different total. Do not silently reinterpret the interval.
A number without its time basis can be misleading even when all arithmetic is correct. Underline “per day,” “per dose” or the stated interval before you choose the first factor. This example does not establish a real medication schedule.
Rounding and notation belong to the final answer
These exercises state their own rounding rules. Keep precision in intermediate steps and round once at the end. An assignment’s rule is not a universal rule for every drug, device or institution.
Write a leading zero for an amount below one (0.4 mL, not .4 mL); avoid an unnecessary trailing zero in medication-style notation (4 mg, not 4.0 mg). Keep a space between number and unit. These presentation checks do not establish that the quantity is appropriate.
Reference: Open RN chapter 5, rounding and decimal notation. For offline practice, use the mixed worksheet and separate worked solutions.
Find the first wrong step
- Wrong final unit: reorient factors before changing arithmetic.
- Right unit, wrong magnitude: check decimals, concentration denominator and time conversion.
- Right number, wrong interval: compare the result’s per-day/per-portion label with the requested quantity.
- Different rounded answer: compare unrounded results before debating the last digit.
- Missing information: identify what must be supplied; do not guess a drop factor, dosing interval or concentration.
When reviewing a mistake, record the setup you used and the exact reason it failed. “I used the wrong formula” is less useful than “I inverted the concentration, so mg did not cancel.”
Continue learning
- Dosage calculation worksheets
- Existing graded dosage practice by skill
- Free NCLEX-RN practice and free NCLEX-PN practice
Original public instruction; no paid question-bank material is reproduced. These pages do not grade your work or predict an exam result.