Dimensional analysis

Dimensional analysis practice for nursing calculations

Practise building the unit path so the unwanted units cancel and only the answer unit survives. Each question is marked on the setup as well as the final number, because in dimensional analysis the setup is the method.

New to the method? Read Medication Calculation Methods Explained

8 practice questions

Write your setup down before checking. Each answer shows the reasoning and the mistake students make most often, because a wrong setup is more useful to understand than a wrong number.

Foundation

You need an answer in mL and you have a dose in mg and a strength of 250 mg in 5 mL. Which way up does the conversion factor go?

Show the setup, answer and why

The unit path

mg × (5 mL / 250 mg)

Answer: Millilitres on top, milligrams underneath, so the milligrams cancel.

The unit you want ends up on top; the unit you are cancelling goes underneath. Writing the factor the other way round leaves mg²/mL, which is a signal the setup is wrong.

A common wrong answer — mg × (250 mg / 5 mL)

Nothing cancels — the milligrams multiply together instead. Checking the units before calculating catches this immediately.

Beginner

500 mg is prescribed from a liquid labelled 250 mg in 5 mL. Set up and solve by dimensional analysis.

Show the setup, answer and why

The unit path

500 mg × (5 mL / 250 mg)

Answer: 10 mL

The milligrams cancel, leaving (500 × 5) ÷ 250 = 10 mL. The surviving unit is mL, which is what the question asked for.

A common wrong answer — 25 000 mg²/mL

The factor was inverted, so nothing cancelled. An answer carrying a squared unit always means the setup, not the arithmetic, needs fixing.

Beginner

A dose of 0.6 g is prescribed from a liquid labelled 200 mg in 5 mL. Set up the full unit path to mL.

Show the setup, answer and why

The unit path

0.6 g × (1000 mg / 1 g) × (5 mL / 200 mg)

Answer: 15 mL

Two factors are chained: grams cancel against the first, milligrams against the second. (0.6 × 1000 × 5) ÷ 200 = 15 mL.

A common wrong answer — 0.015 mL

The gram-to-milligram factor was inverted, dividing by 1000 instead of multiplying. Writing each factor with its units attached makes the direction visible.

Intermediate

An infusion of 1000 mL is to run over 8 hours. Set up the unit path to mL/hr.

Show the setup, answer and why

The unit path

1000 mL / 8 hr

Answer: 125 mL/hr

Only one step is needed. The answer unit mL/hr tells you the volume goes on top and the time underneath.

A common wrong answer — 0.008 hr/mL

The two quantities were placed the wrong way round. Reading the required answer unit first fixes which value belongs on top.

Intermediate

250 mL is to run over 90 minutes. Set up the unit path to mL/hr.

Show the setup, answer and why

The unit path

(250 mL / 90 min) × (60 min / 1 hr)

Answer: Approximately 166.7 mL/hr

The minutes cancel and hours are left underneath, giving (250 × 60) ÷ 90 = 166.666… Rounded at the end, 166.7 mL/hr.

A common wrong answer — 2.8 mL/hr

The minutes-to-hours factor was left out entirely, so the answer is a per-minute rate wearing a per-hour label. The unit path exists precisely to catch this.

Advanced

A supplied order is 5 mg/kg for a patient weighing 60 kg, from a liquid labelled 150 mg in 3 mL. Set up one continuous unit path to mL.

Show the setup, answer and why

The unit path

60 kg × (5 mg / 1 kg) × (3 mL / 150 mg)

Answer: 6 mL

Kilograms cancel against the first factor, milligrams against the second, leaving mL. (60 × 5 × 3) ÷ 150 = 6 mL.

A common wrong answer — Writing the weight factor as (1 kg / 5 mg)

Inverting that factor leaves kg² in the numerator and never reaches milligrams. Each factor has to be oriented so the unwanted unit sits opposite the one it cancels.

Advanced

A supplied order is 3 mcg/kg/min for a patient weighing 70 kg, from a preparation of 400 mcg/mL. Set up the unit path to mL/hr.

Show the setup, answer and why

The unit path

70 kg × (3 mcg / kg·min) × (60 min / 1 hr) × (1 mL / 400 mcg)

Answer: 31.5 mL/hr

Kilograms, minutes and micrograms all cancel in turn, leaving mL/hr. (70 × 3 × 60) ÷ 400 = 31.5 mL/hr.

A common wrong answer — 0.525 mL/hr

The minutes-to-hours factor was omitted, so the result is a per-minute volume. In a chain this long, cancelling the units is the only reliable check that no factor was skipped.

Advanced

A student's setup for a dose in mL ends with an answer expressed in mg. What does that tell you, before you check any arithmetic?

Show the setup, answer and why

The unit path

Check the surviving units, not the number.

Answer: A conversion factor is missing or inverted — the setup never reached mL.

In dimensional analysis the surviving unit is determined entirely by how the factors are arranged. If the answer is not in the unit the question asked for, the arrangement is wrong, whatever the arithmetic says.

A common wrong answer — Assuming the number is right and relabelling it mL

Relabelling changes what the answer claims to be without changing what it actually measures. The unit is the result of the setup, not a caption applied afterwards.

Where to go next

Medication Calculation Methods Explained

How dimensional analysis compares with the formula method and ratio and proportion.

Ratio and proportion practice

The same relationships solved by cross-multiplying instead.

Unit Conversions lesson

The conversion factors most unit paths are built from.

Working Through Multi-Step Calculations

Ordering a longer chain when several factors have to be combined.

8 practice questions

Write your setup down before checking. Each answer shows the reasoning and the mistake students make most often, because a wrong setup is more useful to understand than a wrong number.

Foundation

You need an answer in mL and you have a dose in mg and a strength of 250 mg in 5 mL. Which way up does the conversion factor go?

Show the setup, answer and why

The unit path

mg × (5 mL / 250 mg)

Answer: Millilitres on top, milligrams underneath, so the milligrams cancel.

The unit you want ends up on top; the unit you are cancelling goes underneath. Writing the factor the other way round leaves mg²/mL, which is a signal the setup is wrong.

A common wrong answer — mg × (250 mg / 5 mL)

Nothing cancels — the milligrams multiply together instead. Checking the units before calculating catches this immediately.

Beginner

500 mg is prescribed from a liquid labelled 250 mg in 5 mL. Set up and solve by dimensional analysis.

Show the setup, answer and why

The unit path

500 mg × (5 mL / 250 mg)

Answer: 10 mL

The milligrams cancel, leaving (500 × 5) ÷ 250 = 10 mL. The surviving unit is mL, which is what the question asked for.

A common wrong answer — 25 000 mg²/mL

The factor was inverted, so nothing cancelled. An answer carrying a squared unit always means the setup, not the arithmetic, needs fixing.

Beginner

A dose of 0.6 g is prescribed from a liquid labelled 200 mg in 5 mL. Set up the full unit path to mL.

Show the setup, answer and why

The unit path

0.6 g × (1000 mg / 1 g) × (5 mL / 200 mg)

Answer: 15 mL

Two factors are chained: grams cancel against the first, milligrams against the second. (0.6 × 1000 × 5) ÷ 200 = 15 mL.

A common wrong answer — 0.015 mL

The gram-to-milligram factor was inverted, dividing by 1000 instead of multiplying. Writing each factor with its units attached makes the direction visible.

Intermediate

An infusion of 1000 mL is to run over 8 hours. Set up the unit path to mL/hr.

Show the setup, answer and why

The unit path

1000 mL / 8 hr

Answer: 125 mL/hr

Only one step is needed. The answer unit mL/hr tells you the volume goes on top and the time underneath.

A common wrong answer — 0.008 hr/mL

The two quantities were placed the wrong way round. Reading the required answer unit first fixes which value belongs on top.

Intermediate

250 mL is to run over 90 minutes. Set up the unit path to mL/hr.

Show the setup, answer and why

The unit path

(250 mL / 90 min) × (60 min / 1 hr)

Answer: Approximately 166.7 mL/hr

The minutes cancel and hours are left underneath, giving (250 × 60) ÷ 90 = 166.666… Rounded at the end, 166.7 mL/hr.

A common wrong answer — 2.8 mL/hr

The minutes-to-hours factor was left out entirely, so the answer is a per-minute rate wearing a per-hour label. The unit path exists precisely to catch this.

Advanced

A supplied order is 5 mg/kg for a patient weighing 60 kg, from a liquid labelled 150 mg in 3 mL. Set up one continuous unit path to mL.

Show the setup, answer and why

The unit path

60 kg × (5 mg / 1 kg) × (3 mL / 150 mg)

Answer: 6 mL

Kilograms cancel against the first factor, milligrams against the second, leaving mL. (60 × 5 × 3) ÷ 150 = 6 mL.

A common wrong answer — Writing the weight factor as (1 kg / 5 mg)

Inverting that factor leaves kg² in the numerator and never reaches milligrams. Each factor has to be oriented so the unwanted unit sits opposite the one it cancels.

Advanced

A supplied order is 3 mcg/kg/min for a patient weighing 70 kg, from a preparation of 400 mcg/mL. Set up the unit path to mL/hr.

Show the setup, answer and why

The unit path

70 kg × (3 mcg / kg·min) × (60 min / 1 hr) × (1 mL / 400 mcg)

Answer: 31.5 mL/hr

Kilograms, minutes and micrograms all cancel in turn, leaving mL/hr. (70 × 3 × 60) ÷ 400 = 31.5 mL/hr.

A common wrong answer — 0.525 mL/hr

The minutes-to-hours factor was omitted, so the result is a per-minute volume. In a chain this long, cancelling the units is the only reliable check that no factor was skipped.

Advanced

A student's setup for a dose in mL ends with an answer expressed in mg. What does that tell you, before you check any arithmetic?

Show the setup, answer and why

The unit path

Check the surviving units, not the number.

Answer: A conversion factor is missing or inverted — the setup never reached mL.

A common wrong answer — Assuming the number is right and relabelling it mL

Relabelling changes what the answer claims to be without changing what it actually measures. The unit is the result of the setup, not a caption applied afterwards.