Unit conversions
Also known as: dimensional analysis, unit analysis
Unit conversions change a measurement from one unit to another — such as feet to meters or hours to seconds — by multiplying by conversion factors equal to 1. The systematic method is called dimensional analysis.
A unit conversion rewrites a measurement in different units without changing the quantity itself — turning 3 hours into 180 minutes, or 5 kilometers into 5,000 meters. The tool that makes it systematic is the conversion factor: a fraction equal to 1 because its numerator and denominator represent the same amount, such as 60 minutes / 1 hour or 2.54 cm / 1 inch.
The method, often called dimensional analysis, is to multiply the starting measurement by conversion factors arranged so unwanted units cancel. To convert 45 miles per hour to feet per second: 45 miles/hour × 5,280 feet/mile × 1 hour/3,600 seconds = 66 feet per second. Writing the units into the arithmetic and canceling them like common factors is the built-in error check — if the leftover units are wrong, the setup is wrong.
Two situations deserve extra care. Chained conversions pass through an intermediate unit (inches to feet to yards), multiplying by several factors in sequence. Squared and cubic units require squaring or cubing the entire conversion factor: since 1 yard = 3 feet, 1 square yard = 9 square feet, not 3.
Unit conversions show up across standardized and professional exams. The ACT math and science sections test conversions directly and inside rate problems, the Praxis Core math exam includes them among ratios, proportions, and percents, and the FE Civil exam demands fluent conversion between unit systems in surveying and construction problems. The dimensional analysis habit — write the units, cancel, and check — earns points on all of them.
Key takeaways
- Unit conversions multiply a measurement by conversion factors equal to 1 so the quantity is unchanged.
- Arrange each factor so unwanted units cancel; leftover units reveal setup mistakes.
- Squared and cubic units need the whole conversion factor squared or cubed (1 yd² = 9 ft²).
- Rate conversions like mph to ft/s chain multiple factors in a single expression.
- The ACT, Praxis Core, and FE exams all test unit conversion fluency.