Unit Cancellation

Why Unit Cancellation Over Cross Multiplication?

Cross multiplication works for simple two-ratio problems but breaks down with multi-step calculations. Unit cancellation (dimensional analysis) works for every problem — simple or complex — and produces a visible audit trail of your work.

In this course, unit cancellation is the preferred method for all calculations.

A Note on Cross Multiplication

Cross multiplication is sometimes taught as a shortcut for proportion problems. While it produces correct answers for simple problems, it hides the units and makes errors harder to catch. Unit cancellation is safer, more transparent, and scales to complex problems.

The Core Principle

When the same unit appears in both a numerator and a denominator, it cancels out:

\[\frac{\text{mg}}{\cancel{\text{tablet}}} \times \frac{\cancel{\text{tablet}}}{1} = \text{mg}\]

The unit that does not cancel is your answer unit. If the wrong unit remains, your setup is incorrect.

Which Unit Remains?

The Golden Rule

Before doing any arithmetic, check which unit will remain after cancellation. If it is not the unit you want, flip a conversion fraction and try again.

The unit you want in your answer must not cancel. Every other unit must cancel.

Setting Up the Chain

Write your given value first, then attach conversion fractions so unwanted units cancel one by one:

\[\text{Given value} \times \frac{\text{wanted unit}}{\text{unwanted unit}} = \text{answer in wanted unit}\]

Which Unit Remains — Practice

These problems focus entirely on identifying the surviving unit before doing any arithmetic. This builds the habit of checking your setup before calculating.

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Problem 1 — Which unit remains?

\[\frac{500 \text{ mg}}{1} \times \frac{1 \text{ tablet}}{250 \text{ mg}}\]

Answer

mg appears in the numerator of the first fraction and the denominator of the second — it cancels.

tablet appears only in the numerator — it remains.

\[\frac{500 \cancel{\text{ mg}}}{1} \times \frac{1 \text{ tablet}}{250 \cancel{\text{ mg}}} = \frac{500}{250} \text{ tablets} = 2 \text{ tablets}\]

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Problem 2 — Which unit remains?

\[\frac{2.5 \text{ g}}{1} \times \frac{1000 \text{ mg}}{1 \text{ g}}\]

Answer

g cancels. mg remains.

\[\frac{2.5 \cancel{\text{ g}}}{1} \times \frac{1000 \text{ mg}}{1 \cancel{\text{ g}}} = 2500 \text{ mg}\]

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Problem 3 — Which unit remains?

\[\frac{176 \text{ lb}}{1} \times \frac{1 \text{ kg}}{2.2 \text{ lb}} \times \frac{5 \text{ mg}}{1 \text{ kg}}\]

Answer

lb cancels with lb. kg cancels with kg. mg remains.

\[\frac{176 \cancel{\text{ lb}}}{1} \times \frac{1 \cancel{\text{ kg}}}{2.2 \cancel{\text{ lb}}} \times \frac{5 \text{ mg}}{1 \cancel{\text{ kg}}} = \frac{880}{2.2} \text{ mg} = 400 \text{ mg}\]

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Problem 4 — Which unit remains?

\[\frac{3 \text{ tsp}}{1} \times \frac{5 \text{ mL}}{1 \text{ tsp}} \times \frac{1 \text{ L}}{1000 \text{ mL}}\]

Answer

tsp cancels. mL cancels. L remains.

\[\frac{3 \cancel{\text{ tsp}}}{1} \times \frac{5 \cancel{\text{ mL}}}{1 \cancel{\text{ tsp}}} \times \frac{1 \text{ L}}{1000 \cancel{\text{ mL}}} = \frac{15}{1000} \text{ L} = 0.015 \text{ L}\]

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Problem 5 — Spot the Error

A student sets up this calculation to find how many tablets to give for a 500 mg order with 250 mg per tablet. What is wrong?

\[\frac{500 \text{ mg}}{1} \times \frac{250 \text{ mg}}{1 \text{ tablet}}\]

Answer

mg does not cancel — it appears in the numerator of both fractions. The surviving unit would be mg²/tablet which is meaningless.

The conversion fraction is upside down. Correct setup:

\[\frac{500 \cancel{\text{ mg}}}{1} \times \frac{1 \text{ tablet}}{250 \cancel{\text{ mg}}} = 2 \text{ tablets}\]

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Full Calculation Practice

Problem 6

Order: 750 mg. Stock: 250 mg/5 mL. How many mL?

Set up the unit cancellation chain first, identify the surviving unit, then calculate.

Answer

Setup: [\frac{750 \cancel{\text{ mg}}}{1} \times \frac{5 \text{ mL}}{250 \cancel{\text{ mg}}}]

mg cancels. mL remains. ✅

\[= \frac{750 \times 5}{250} \text{ mL} = \frac{3750}{250} = 15 \text{ mL}\]

Problem 7

Order: 0.5 g. Stock: 250 mg per tablet. How many tablets?

Note: units don't match — conversion required.

Answer

Setup: [\frac{0.5 \cancel{\text{ g}}}{1} \times \frac{1000 \cancel{\text{ mg}}}{1 \cancel{\text{ g}}} \times \frac{1 \text{ tablet}}{250 \cancel{\text{ mg}}}]

g cancels. mg cancels. tablet remains. ✅

\[= \frac{0.5 \times 1000}{250} \text{ tablets} = \frac{500}{250} = 2 \text{ tablets}\]

If Your Units Don't Cancel Cleanly

Stop. Do not calculate. Recheck your conversion fractions. A setup with non-cancelling units will always produce a wrong answer regardless of the arithmetic.