ACT® · Math: Integrating Essential Skills
Measurement Units and Unit Conversion
Unit conversion works by multiplying by fractions equal to 1, built so the unwanted unit cancels and the wanted unit survives. For area and volume, the length factor must be squared or cubed, not just multiplied.
A conversion factor like 1 mile / 5280 feet equals 1, since a mile and 5280 feet are the same distance. Multiplying by it never changes the actual quantity, only how it is labeled. Write it with the unit you want to kill on the bottom and the unit you want on top, and that bottom unit cancels against whatever unit sits on top of your starting number. Chain several of these fractions in a row and the cancellation runs diagonally: each unlucky unit on top of one fraction meets its match on the bottom of the next, until the only unit left standing is the one you need.
This matters most once you leave simple length. Area is length times length, so converting square yards to square feet is not a single multiplication by 3. Each yard of the two dimensions turns into 3 feet, so the factor gets squared: 3 x 3 = 9 square feet per square yard. Volume pushes this further: a cubic yard is 3 feet by 3 feet by 3 feet, so the factor is cubed: 27 cubic feet per cubic yard. Forgetting to square or cube the factor, and just scaling by 3, is the single most common error on these problems.
This sits right next to rate problems like miles per hour, where you chain two or three conversion fractions at once: feet per second into miles per hour needs both a distance conversion and a time conversion stacked together. Same tool, just more links in the chain.
Key terms
- conversion factor
- A fraction equal to 1 built from two equal quantities in different units, used to relabel a measurement without changing its size.
- dimensional analysis
- The method of chaining conversion fractions so units cancel and only the desired unit remains.
- squared factor
- In area conversions, the linear conversion number must be multiplied by itself since area is length times length.
- cubed factor
- In volume conversions, the linear conversion number must be multiplied by itself three times since volume is length times length times length.
Watch it explained
- How it worksHow does a chain of conversion fractions cancel diagonally until only the unit you want survives?
- Worked exampleFind the speed in miles per hour of a runner moving at 22 feet per second.
- Worked exampleFind how many cubic feet fit in 2 cubic yards, and why the answer is not 6.
- Going furtherWhat happens to an area conversion when the length conversion factor has to be squared?
Check yourself
End of topic.